An Exercise in Probability Magic
A version of this article appears in ‘Twisted Logic: Puzzles, Paradoxes, and Big Questions’, by Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.
SIZE MATTERS
What is the minimum number of individuals that need to be present in the room for it to be more likely than not that at least two of them share a birthday? This is what the ‘Birthday Paradox’ (or ‘Birthday Problem’) seeks to solve.
For the sake of simplicity, let’s assume that all calendar dates have an equal chance of being someone’s birthday and let’s disregard the Leap Year occurrence of 29 February.
A BASIC INTUITION: ANALYSING THE ODDS
At first glance, you might think that the odds of two people sharing a birthday are incredibly low. In a group of just two people, the likelihood of them sharing a birthday is a mere 1/365. Why is that? We have 365 days in a year, hence there’s only one chance in 365 that the second person would have been born on the same specific day as the first person.
Now, let’s take a group of 366 people. In this case, it’s certain that at least one person shares a birthday with someone else, due to the simple fact that we only have 365 possible birthdays (ignoring Leap Years).
The initial intuition may suggest that the tipping point—the group size at which there’s a 50% chance of two individuals sharing a birthday—is around the midpoint of these two extremes. You may think it lies around a group size of about 180. However, the reality is surprisingly different, and the actual answer is much smaller.
THE CALCULATIONS: UNRAVELLING THE BIRTHDAY PARADOX
To understand the concept better, we need to dig deeper into the probabilities involved. Let’s consider a duo: Julia and Julian. Let’s assume that Julia’s birthday falls on 1 May. The chance that Julian shares the same birthday, assuming an equal distribution of birthdays across the year, is 1/365.
What about the probability that Julian doesn’t share a birthday with Julia? It’s simply 1 minus 1/365, or 364/365. This number illustrates the chance that the second person in a random duo has a different birthday than the first person.
Adding a third person into the mix changes things slightly. The chance that all three birthdays are different is the chance that the first two are different (364/365) multiplied by the probability that the third birthday is unique (363/365). So, the probability of three different birthdays equals (364/365) × (363/365).
As we expand the group, the calculations continue in a similar manner. The more people in the room, the greater the chance of finding at least two people sharing a birthday.
Consider a group of four people. The probability that four people have different birthdays is (364 × 363 × 362)/(365 × 365 × 365). To find the probability that at least two of the four share a birthday, we subtract this number from 1. Thus, the odds of having at least two people with the same birthday in a group of four are about 1.6%.
As the number of people in the room increases, the probability of at least two sharing a birthday grows:
With 5 people, the probability is 2.7%.
With 10 people, the probability is 11.7%.
With 16 people, the probability is 28.1%.
With 23 people, the probability is 50.5%.
With 32 people, the probability is 75.4%.
With 40 people, the probability is 89.2%.
THE PARADOX UNVEILED: IT’S NOT JUST ABOUT BIRTHDAYS
You might be wondering why we need just 23 people to reach a 50% chance of finding shared birthdays. This can be explained by how many possible pairs can be made in a group. In a group of 23, there are 253 unique pairs. Each of these pairs has a 1/365 chance of sharing a birthday, and all these possibilities add up. This is what makes the birthday problem so counterintuitive. Basically, when a large group is analysed, there are so many potential pairings that it becomes statistically likely for coincidental matches to occur.
This is a perfect demonstration of the concept of multiple comparisons and an example of the so-called ‘Multiple Comparisons Fallacy’.
The same reasoning applies to balls being randomly dropped into open boxes. Assume there is an equal chance that a ball will drop into any of the individual boxes, and there are 365 such boxes, into which 23 balls are randomly dropped. There is an an equal chance, we assume, that a ball will drop into any specific box. Now, there is just over a 50% chance in this scenario that there will be at least two balls in at least one of the boxes. Randomness produces more aggregation than intuition leads us to expect.
YOUR PERSONAL BIRTHDAY CHANCES: WHERE DO YOU STAND?
The reason for the paradox is that the question is not asking about the chance that someone shares your particular birthday or any particular birthday. It is asking whether any two people share any birthday.
While the birthday problem shows the increased likelihood of shared birthdays in a group, the chance that someone shares your birthday specifically is a different question.
In a group of 23 people, including yourself, the probability that at least one person shares your birthday is much lower than 50%—it’s about 6%. This is because there are only 22 potential pairings that include you.
Even in a group of 366 people, the probability that someone shares your specific birthday is only around 63%.
CONCLUSION: THE MAGIC OF PROBABILITY AND THE BIRTHDAY PARADOX
The Birthday Paradox reveals an intriguing counterintuitive fact about probability: a group of just 23 people has a greater than 50% chance of including at least two people who share the same birthday. It sheds light on the intricacies of probability by demonstrating how many opportunities there are for matches to occur, even in seemingly small groups. For example, if you can find out the birthdays of the 22 players at the start of a football game, and the referee, more than half of the time two of them will share a birthday.
This fascinating concept has applications way beyond birthdays. It’s also very important for the safety and performance of computer systems and online security. This idea helps specialists prevent and deal with issues that occur when data unexpectedly overlaps. Understanding the paradox is crucial, therefore, for those who design and secure computer systems, helping them to make these systems more reliable and efficient.
Nevertheless, it’s in the social setting of parties where the paradox becomes a delightful surprise. Next time you’re among friends or at any casual meet-up, consider introducing this paradox; you might just bring to life the unexpected magic of probability!
The Timeless Bayesian
When Should We Trust a Loved One? Exploring a Shakespearean Tragedy
OTHELLO: THE BACKGROUND
Created by William Shakespeare, ‘Othello’ is a play centred around four main characters: Othello, a general in the Venetian army; his devoted wife, Desdemona; his trusted lieutenant, Cassio; and his manipulative ensign, Iago. Iago’s plan forms the central conflict of the play. Driven by jealousy and a large helping of evil, Iago seeks to convince Othello that Desdemona is conducting a secret affair with Cassio. His strategy hinges on a treasured keepsake, a precious handkerchief which Desdemona received as a gift from Othello. Iago conspires successfully to plant this keepsake in Cassio’s lodgings so that Othello will later find it.
UNDERSTANDING OTHELLO’S MINDSET
Othello’s reaction to this discovery can potentially take different paths, depending on his character and mindset. If Othello refuses to entertain any possibility that Desdemona is being unfaithful to him, then no amount of evidence could ever change that belief.
On the other hand, Othello might accept that there is a possibility, however small, that Desdemona is being unfaithful to him. This would mean that there might be some level of evidence, however overwhelming it may need to be, that could undermine his faith in Desdemona’s loyalty.
There is, however, another path that Othello could take, which is to evaluate the circumstances objectively and analytically, weighing the evidence. But this balanced approach also has its pitfalls. A very simple starting assumption that he could make would be to assume that the likelihood of her guilt is equal to the likelihood of her innocence. That would mean assigning an implicit 50% chance that Desdemona had been unfaithful. This is known as the ‘Prior Indifference Fallacy’. If the prior probability is 50%, this needs to be established by a process better than simply assuming that because there are two possibilities (guilty or innocent), we can ascribe automatic equal weight to each. If Othello falls into this trap, any evidence against Desdemona starts to become very damning.
THE LOGICAL CONTRADICTION APPROACH
An alternative approach would be to seek evidence that directly contradicts the hypothesis of Desdemona’s guilt. If Othello could find proof that logically undermines the idea of her infidelity, he would have a solid base to stand on. However, there is no such clear-cut evidence, leading Othello deeper into a mindset of anger and suspicion.
BAYES’ THEOREM TO THE RESCUE
Othello might seek a strategy that allows him to combine his subjective belief with the new evidence to form a rational judgement. This is where Bayes’ theorem comes in. Bayes’ theorem allows, as we have seen in previous chapters, for the updating of probabilities based on observed evidence. The theorem can be expressed in the following formula:
Updated probability = ab/[ab + c (1 − a)]
In this formula, a is the prior probability, representing the likelihood that a hypothesis is true before encountering new evidence. b is the conditional probability, describing the likelihood of observing the new evidence if the hypothesis is true. And finally, c is the probability of observing the new evidence if the hypothesis is false. In this case, the evidence is the keepsake in Cassio’s lodgings, and the hypothesis is that Desdemona is being unfaithful to Othello.
APPLYING BAYES’ THEOREM TO OTHELLO’S DILEMMA
Now, before he discovers the keepsake (new evidence), suppose Othello perceives a 4% chance of Desdemona’s infidelity (a = 0.04). This represents his prior belief, based on his understanding of Desdemona’s character and their relationship. Of course, he is not literally assigning percentages, but he is doing so implicitly, and here we are simply making these explicit to show what might be happening within a Bayesian framework.
Next, consider the probability of finding the keepsake in Cassio’s room if Desdemona is indeed having an affair. Let’s assume that Othello considers there is a 50% chance of this being the case (b = 0.5).
Finally, what is the chance of finding the keepsake in Cassio’s room if Desdemona is innocent? This would in Othello’s mind require an unlikely series of events, such as the handkerchief being stolen or misplaced, and then ending up in Cassio’s possession. Let’s say he assigns this a low probability of just 5% (c = 0.05).
BAYESIAN PROBABILITIES: WEIGHING THE EVIDENCE
Feeding these values into Bayes’ equation, we can calculate the updated (or posterior) probability of Desdemona’s guilt in Othello’s eyes, given the discovery of the keepsake. The resulting probability comes out to be 0.294 or 29.4%. This suggests that, after considering the new evidence, Othello might reasonably believe that there is nearly a 30% chance that Desdemona is being unfaithful.
IAGO’S MANIPULATION OF PROBABILITIES
This 30% likelihood might not be high enough for Iago’s deceitful purposes. To enhance his plot, Iago needs to convince Othello to revise his estimate of c downwards, arguing that the keepsake’s presence in Cassio’s room is a near-certain indication of guilt. If Othello lowers his estimate of c from 0.05 to 0.01, the revised Bayesian probability shoots up to 67.6%. This change dramatically amplifies the perceived impact of the evidence, making Desdemona’s guilt appear significantly more probable.
DESDEMONA’S DEFENCE STRATEGY
On the other hand, Desdemona’s strategy for defending herself could be to challenge Othello’s assumption about b. She could argue that it would be illogical for her to risk the discovery of the keepsake if she were truly having an affair with Cassio. By reducing Othello’s estimate of b, she can turn the tables and make the presence of the keepsake testimony to her innocence rather than guilt.
CONCLUSION: THE TIMELESS BAYESIAN
Shakespeare’s ‘Othello’ was written about a century before Thomas Bayes was born. Yet the complex interplay of trust, deception, and evidence in the tragedy presents a classic case study in Bayesian reasoning.
Shakespeare was inherently Bayesian in his thinking. The tragedy of the play is that Othello was not!
Exploring Bayes’ Theorem Through a Story
A version of this article appears in Twisted Logic: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.
UNDERSTANDING EVIDENCE THROUGH A STORY
Imagine a situation where your friend, known for her upstanding character, is accused of vandalising a shop window. The only evidence against her is a police officer’s identification. You know her well and find it hard to believe she would commit such an act.
SETTING THE STAGE FOR BAYESIAN ANALYSIS
In Bayesian terms, the ‘condition’ is your friend being accused, while the ‘hypothesis’ is that she’s guilty. To apply Bayes’ theorem, we consider three probabilities:
Prior Probability: Based on your knowledge of your friend, you might initially think there’s a low chance of her guilt, say 5%. This is your ‘prior’—your belief before considering the new evidence.
Likelihood of Evidence If Guilty: Consider how reliable the officer’s identification is. If your friend were guilty, what’s the chance that the officer would identify her correctly? Let’s estimate this at 80%.
Likelihood of Evidence If Innocent: What’s the chance of the officer mistakenly identifying your friend if she’s innocent? Factors like similar appearances or biases could play a role. Let’s estimate this at 15%.
THE ITERATIVE NATURE OF BAYESIAN UPDATING
Bayes’ theorem allows for continual updates. If new evidence arises, you can recalculate, using your updated belief as the new ‘prior’. This process offers a dynamic way to assess situations as they evolve.
WHEN EVIDENCE DOESN’T ADD UP
In cases where evidence is equally likely whether the hypothesis is true or false, it doesn’t change our belief. It’s crucial to evaluate the quality of evidence, not just its existence.
CHALLENGES IN ASSIGNING PROBABILITIES
While assigning precise probabilities to real-life situations can be challenging, the exercise is invaluable. It forces us to think critically and systematically about our beliefs and how new information affects them.
The Unfolding Story
Now let’s consider the story in a little more detail. You’ve received a phone call from your local police station. An officer tells you that your friend, someone you’ve known for years, is currently assisting the police in their investigation into a case of vandalism. The crime in question involves a shop window that was smashed on a quiet street, close to where she resides. Furthermore, the incident took place at noon that day, which happens to be her day off work.
You had heard about the incident, but had no reason to believe your friend was involved. After all, she’s not a person known for reckless or unlawful behaviour.
However, this is where the narrative takes a twist. Your friend comes to the phone and tells you that she’s been charged with the crime. The accusation primarily stems from the assertion of a police officer who has positively identified her as the offender. There’s no other evidence, such as CCTV footage or eyewitness testimonies, to substantiate the officer’s claim.
She vehemently maintains her innocence, insisting it’s a case of mistaken identity.
The Challenge
Now, as a follower of Bayes as well as being a close friend, you find yourself in a position where you need to evaluate the probability that she has committed the crime before deciding how to advise her. This challenge leads us to the central theme of this section—the application of Bayes’ theorem to real-life situations.
Before we proceed, let’s clarify our terms. The ‘condition’ in this context is that your friend has been accused of causing the criminal damage. The ‘hypothesis’ we aim to assess is the probability that she is indeed guilty.
Bayes’ Theorem and Its Application
So, how does Bayes’ theorem help us answer this question? Well, Bayes’ theorem is a formula that describes how to update the probabilities of hypotheses being true when given new evidence. It follows the logic of probability theory, adjusting initial beliefs based on the weight of evidence.
To apply Bayes’ theorem, we need to estimate three crucial probabilities:
Prior probability (‘a’)
The prior probability refers to the initial assessment of the hypothesis being true, independent of the new evidence. In this scenario, it equates to the likelihood you assign to your friend being guilty before you hear the evidence.
Considering you’ve known her for years and her involvement in such an act is uncharacteristic, you might deem this probability low. After a thoughtful consideration of your friend’s past actions and character, allowing for the fact that she was off work on that day and in the neighbourhood, let’s say you assign a 5% chance (0.05) to her being guilty.
Assigning this prior probability requires an honest evaluation of your initial beliefs, unaffected by the newly received information.
Conditional probability of evidence given hypothesis is true (‘b’)
Next, you need to estimate the likelihood that the new evidence (officer’s identification) would have arisen if your friend were indeed guilty.
This estimate might be guided by factors such as the officer’s reliability, credibility, and proximity to the crime scene. For the sake of argument, let’s estimate this probability to be 80% (0.8).
Conditional probability of evidence given hypothesis is false (‘c’)
The third estimate involves figuring out the probability that the new evidence would surface if your friend is innocent. This entails gauging the chance that the officer identifies your friend as the offender when she isn’t guilty.
The probability could be influenced by several factors—perhaps the officer saw someone of similar age and appearance, jumped to conclusions, or has other motivations. For the purposes of our discussion, let’s estimate this probability to be 15% (0.15).
Probabilities Adding Up
An interesting point to note is that the sum of probabilities ‘b’ and ‘c’ doesn’t necessarily have to equal 1. Just for example, the police officer might have a reason to identify your friend either way (whether she’s guilty or innocent), in which case the sum of ‘b’ and ‘c’ could exceed 1. Alternatively, the officer may be reluctant to positively identify a suspect in any circumstance unless he is absolutely certain; in which case b plus c may well sum to rather less than 1. In this particular narrative, b plus c add up to 0.95.
Calculation and Interpretation
With these estimates in hand, we can now apply Bayes’ theorem, which calculates the posterior probability (the updated probability of the hypothesis being true after considering new evidence) using the formula: ab/[ab + c (1 − a)].
In our case, substituting the values results in a posterior probability of around 21.9%. What does this mean? Despite the officer’s confident identification (a seemingly strong piece of evidence), there’s only a 21.9% probability that your friend is guilty given the current information.
This result may seem counterintuitive. However, this discrepancy arises from our understanding of prior probability and the weight we assign to the new evidence. We must remember that the officer’s identification is only one piece of the puzzle, and its strength as evidence is balanced against the prior probability and the potential for a false identification.
Updating the Probability
The beauty of Bayes’ theorem lies in its iterative nature. Let’s suppose that another piece of evidence emerges—say, a second witness identifies your friend as the culprit. You can reapply Bayes’ theorem, using the posterior probability from the previous calculation as the new prior probability. This iterative process allows you to incorporate additional pieces of evidence, each of which updates the probability you assign to your friend’s guilt or innocence.
Cases Where Evidence Adds No Value
Consider a situation where ‘b’ equals 1 and ‘c’ also equals 1. This would imply that the officer would identify your friend as guilty whether she was or not. In such cases, the identification fails to update the prior probability, and the posterior probability remains the same as the initial prior probability.
The Imperfections of Assigning Probabilities
Now, it’s worth recognising the potential difficulty in assigning precise probabilities to real-life situations. After all, our scenario involves complex human behaviour and a unique event.
However, our inability to determine precise probabilities shouldn’t lead us to dismiss the process. In fact, this process of estimation is what we’re doing implicitly when we evaluate situations in our everyday lives.
While the results might not be perfect, Bayes’ theorem provides a systematic approach to updating our beliefs in the face of new evidence.
CONCLUSION: BAYESIAN REASONING IN REAL LIFE
Bayes’ theorem provides a structured approach to incorporating new evidence into our beliefs. It’s a tool that enhances our decision-making, offering a mathematical framework to navigate uncertainties, from everyday dilemmas to complex legal and medical decisions.
As we grapple with uncertainty, the application of Bayes’ theorem allows us to transition from ignorance to knowledge, systematically and rationally. Thus, whether we’re faced with a shattered shop window or any other challenging situation, we have a powerful tool to help us navigate our path towards truth.
Exploring a Courtroom Tragedy
A version of this article appears in Twisted Logic: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams. Chapman and Hall/CRC Press, 2024.
THE CONVICTION
In the final weeks of the 20th century, a lawyer named Sally Clark was convicted of the murder of her two infant sons. Despite being a woman of good standing with no history of violent behaviour, Clark was swept up in a whirlwind of accusations, trials, and appeals that would besmirch the criminal justice system and cost her dearly.
THE INVESTIGATION AND TRIAL—BUILDING A CASE ON UNCERTAINTY
The deaths of Clark’s two children were initially assumed to be tragic instances of Sudden Infant Death Syndrome (SIDS), a cause of infant mortality that was not well understood even by medical experts. However, the authorities became suspicious of the coincidental deaths, leading to Clark’s eventual trial. As the investigation evolved, it subsequently transpired that numerous pieces of evidence helpful to the defence were withheld from them.
STATISTICAL EVIDENCE—THE MISINTERPRETATION
The prosecution presented a piece of seemingly damning statistical evidence during Clark’s trial. One of their witnesses, a paediatrician, asserted that the probability of two infants from the same family dying from SIDS was incredibly low—approximately 1 in 73 million. He compared the odds to winning a bet on a longshot in the iconic Grand National horse race four years in a row.
THE PROSECUTOR’S FALLACY—THE DANGEROUS CONFLATION OF PROBABILITIES
The flaws in the statistical argument presented at the trial were both substantial and consequential. The paediatrician had mistakenly assumed that the deaths of Clark’s children were unrelated, or ‘independent’ events. This assumption neglects the potential for an underlying familial or genetic factor that might contribute to SIDS.
Moreover, the paediatrician’s argument represents a common misinterpretation of probability known as the ‘Prosecutor’s Fallacy’. This fallacy involves conflating the probability of observing specific evidence if a hypothesis is true, with the probability that the hypothesis is true given that evidence. These are two very different things but easy for a jury of laymen to confuse.
THE PROSECUTOR’S FALLACY EXPLAINED
This fallacy arises from confusing two different probabilities:
The probability of observing specific evidence (in this case, two SIDS deaths) if a hypothesis (Clark’s guilt) is true.
The probability that the hypothesis is true given the observed evidence.
THE NEED FOR COMPARATIVE LIKELIHOOD ASSESSMENT
The Royal Statistical Society emphasised the need to compare the likelihood of the deaths under each hypothesis—SIDS or murder. The rarity of two SIDS deaths alone doesn’t provide sufficient grounds for a murder conviction.
PRIOR PROBABILITY—UNDERSTANDING THE LIKELIHOOD OF GUILT BEFORE THE EVIDENCE
Prior probability—a concept integral to understanding the Prosecutor’s Fallacy—is often overlooked in court proceedings. This term refers to the probability of a hypothesis (in this case, that Sally Clark is a child killer) being true before any evidence is presented.
Given that she had no history of violence or harm towards her children, or anyone else, or any indication of such a tendency, the prior probability of her being a murderer would be extremely low. In fact, the occurrence of two cases of SIDS in a single family is much more common than a mother murdering her two children.
The jury should weigh up the relative likelihood of the two competing explanations for the deaths. Which is more likely? Double infant murder by a mother or double SIDS?
More generally, it is likely in any large enough population that one or more cases of something highly improbable will occur in any particular case.
In a letter from the President of the Royal Statistical Society to the Lord Chancellor, Professor Peter Green explained the issue succinctly:
The jury needs to weigh up two competing explanations for the babies’ deaths: SIDS or murder. The fact that two deaths by SIDS is quite unlikely is, taken alone, of little value. Two deaths by murder may well be even more unlikely. What matters is the relative likelihood of the deaths under each explanation, not just how unlikely they are under one explanation.
Put another way, before considering the evidence, the prior probability of Clark being a murderer, given her background and lack of violent history, was extremely low. The probability of two SIDS deaths in one family, while rare, was still much higher than the likelihood of the mother murdering her two children.
The jury should weigh up the relative likelihood of the two competing explanations for the deaths. Which is more likely? Double infant murder by a mother or double SIDS?
More generally, it is likely in any large enough population that one or more cases of something highly improbable will occur in any particular case.
In a letter from the President of the Royal Statistical Society to the Lord Chancellor, Professor Peter Green explained the issue succinctly:
The jury needs to weigh up two competing explanations for the babies’ deaths: SIDS or murder. The fact that two deaths by SIDS is quite unlikely is, taken alone, of little value. Two deaths by murder may well be even more unlikely. What matters is the relative likelihood of the deaths under each explanation, not just how unlikely they are under one explanation.
Put another way, before considering the evidence, the prior probability of Clark being a murderer, given her background and lack of violent history, was extremely low. The probability of two SIDS deaths in one family, while rare, was still much higher than the likelihood of the mother murdering her two children.
THE NEED FOR COMPARATIVE LIKELIHOOD ASSESSMENT
The Royal Statistical Society emphasised the need to compare the likelihood of the deaths under each hypothesis—SIDS or murder. The rarity of two SIDS deaths alone doesn’t provide sufficient grounds for a murder conviction.
The Fictional Case of Lottie Jones
To illustrate the Prosecutor’s Fallacy, consider the fictional case of Lottie Jones, charged with winning the lottery by cheating. The fallacy occurs when the expert witness equates the low probability of winning the lottery (1 in 45 million) with the probability that a lottery win was achieved unfairly.
As in the Sally Clark case, the prosecution witness in this fictional parody commits the classic ‘Prosecutor’s Fallacy’. He assumes that the probability Lottie is innocent of cheating, given that she won the Lottery, is the same thing as the probability of her winning the Lottery if she is innocent of cheating. The former probability is astronomically higher than the latter unless we have some other indication that Lottie has cheated to win the Lottery. It is a clear example of how it is likely, in any large enough population, that things will happen that are improbable in any particular case. In other words, the 1 in 45 million represents the probability that a Lottery entry at random will win the jackpot, not the probability that a player who has won did so fairly!
Lottie just got very, very lucky just as Sally Clark got very, very unlucky.
THE AFTERMATH—TRAGEDY AND LESSONS LEARNED
Following her acquittal in 2003, Sally Clark never recovered from her ordeal and sadly died just a few years later. Her story stands as testament to the potential for disastrous consequences when statistics are misunderstood or misrepresented.
O.J. SIMPSON—AN ALTERNATE SCENARIO
Even in high-profile cases, such as American former actor and NFL football star O.J. Simpson’s murder trial in the 1990s, this same misinterpretation of statistics is prevalent. Simpson’s defence team argued that it was unlikely Simpson killed his wife because only a small percentage of spousal abuse cases result in the spouse’s death. This argument, though statistically accurate, overlooks the relevant information—the fact that about 1 in 3 murdered women were killed by a spouse or partner. This represents a very clear case of the misuse of the Inverse or Prosecutor’s Fallacy in argumentation before a jury.
CONCLUSION: THE IMPORTANCE OF STATISTICAL LITERACY
The importance of statistics in our justice system cannot be overstated. We must recognise the potential for misinterpretation and the potentially devastating results. A concerted effort to promote statistical literacy, particularly within our legal systems, can hopefully go a long way in preventing future miscarriages of justice.
Exploring the World of False Positives
A version of this article appears in Twisted Logic: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.
THE FLU TEST SCENARIO: SETTING THE STAGE
Imagine this scenario: you twist your knee in a skateboarding mishap and decide to visit your doctor to have it looked at, just to be on the safe side. At the surgery, they run a routine test for a flu virus on all their patients, based on the estimate that about 1 out of every 100 patients visiting them will have the virus. This flu test is known to be pretty accurate—it gets the diagnosis right 99 out of 100 times. In other words, it correctly identifies 99% of people who are sick as sick, and equally importantly, it correctly clears 99% of those who don’t have the flu virus.
Now, you take the test, and to your surprise, it comes back positive. What does this mean for you, exactly? You dropped in to have your knee looked at, and now it seems you have the flu.
To summarise the situation, imagine you’ve twisted your knee and, while at the doctor’s office, you’re given a routine flu test. The test is 99% accurate and is positive. But what are the actual chances that you have the flu? This scenario is perfect for exploring Bayes’ theorem and understanding false positives.
BREAKING DOWN THE INVERSE FALLACY
Here, we step into the tricky territory of probabilities, a place where common sense can often mislead us. So, what is the chance that you do have the virus?
The intuitive answer is 99%, as the test is 99% accurate. But is that right?
The information we are given relates to the probability of testing positive given that you have the virus. What we want to know, however, is the probability of having the virus given that you test positive. This is a crucial difference.
Common intuition conflates these two probabilities, but they are very different. If the test is 99% accurate, this means that 99% of those with the virus test positive. But this is NOT the same thing as saying that 99% of patients who test positive have the virus. This is an example of the ‘Inverse Fallacy’ or ‘Prosecutor’s Fallacy’. In fact, those two probabilities can diverge markedly.
To summarise, common sense might suggest a 99% chance of having the flu, aligning with the test’s accuracy. However, this confuses the probability of testing positive when having the flu with the probability of having the flu when testing positive—a common mistake known as the ‘Inverse Fallacy’.
So what is the probability you have the virus if you test positive, given that the test is 99% accurate? To answer this, we can use Bayes’ theorem.
APPLYING BAYES’ THEOREM
Bayes’ theorem, as we have seen, uses three values:
Your initial chance of having the flu before taking the test, which in our scenario was estimated to be 1 out of 100 or 0.01.
The likelihood of the test showing a positive result if you have the flu, which we know to be 99% or 0.99 based on the accuracy of the test.
The likelihood of the test showing a positive result if you don’t have the flu, which is 1% or 0.01, again based on the accuracy of the test.
When we plug these into Bayesian formula, we end up with a surprising result. If you test positive for the flu, despite the test being 99% accurate, there’s actually only a 50% chance that you really have it.
In other words, to find the real probability of having the flu, we consider:
Prior Probability: Your initial chance of having the flu is 1% (1 in 100).
True Positive Rate: The test correctly identifies the flu 99% of the time.
False Positive Rate: The test incorrectly indicates flu in healthy individuals 1% of the time.
The formula is expressed as follows:
ab/[ab + c (1 − a)]
where
a is the prior probability, i.e. 0.01,
b is 0.99.
c is 0.01.
Using Bayes’ theorem, we find a surprising result: even with a 99% accurate test, there’s only a 50% chance you have the flu after a positive result.
GRAPPLING WITH PROBABILITIES
The result can seem counterintuitive, and it’s worth taking a moment to understand why that is. The key is to remember that the flu is a relatively rare occurrence—only 1 in 100 patients have it. While the test may be 99% accurate, we have to take into account the relative rarity of the disease in those who are tested. The chance is just 1 in 100. The chance of having the flu before taking the test and the chance of the test making an error are both, therefore, 1 in 100. These two probabilities are the same, and so, when you test positive, the chance that you have the flu is actually just 1 in 2.
It is basically a competition between how rare the virus is and how rarely the test is wrong. In this case, there is a 1 in 100 chance that you have the virus before taking the test, and the test is wrong one time in 100. These two probabilities are equal, so the chance that you have the virus when testing positive is 1 in 2, despite the test being 99% accurate.
Put another way, the counterintuitive outcome arises because the flu is relatively rare (1 in 100), balancing against the test’s accuracy.
THE IMPLICATION OF SYMPTOMS AND PRIOR PROBABILITIES
This calculation changes if we add in some more information. Let’s say you were already feeling unwell with flu-like symptoms before the test. In this case, your doctor might think you’re more likely to have the flu than the average patient, and this would increase your ‘prior probability’. Consequently, a positive test in this context would be more indicative of actually having the flu, as it aligns with both the symptoms and the test result.
In this way, Bayes’ theorem incorporates both the statistical likelihood and real-world information. It’s a powerful tool to help us understand probabilities better and to make informed decisions. The bottom line, though, is that while a positive test result can be misinterpreted, it should, especially in conjunction with symptoms, be taken seriously.
The Role of Symptoms in Adjusting Probabilities
If you had flu-like symptoms before the test, this would increase your ‘prior probability’. Consequently, a positive test in this context would be more indicative of actually having the flu, as it aligns with both the symptoms and the test result.
CONCLUSION: THE BROAD APPLICATION OF BAYESIAN THINKING
While we’ve used the example of a flu test, the principles of Bayes’ theorem apply beyond the doctor’s door. From the courtroom to the boardroom, from deciding if an email is spam to weighing up the reliability of a rumour, we often need to update our beliefs in the face of new evidence. Remember, a single piece of evidence should always be weighed against the broader context and initial probabilities.
Exploring the Texas Sharpshooter Fallacy
Further discussion of the flawed use of statistics in the Courtroom is available in Twisted Logic: Puzzles, Paradoxes, and Big Questions, by Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024, and also in Probability, Choice, and Reason, by the same author and publisher,
The Texas Sharpshooter Fallacy and the Lucy Letby Case: A Statistical Illusion?
The Texas Sharpshooter Fallacy is a cognitive bias where patterns are imposed on random data after the fact, creating the illusion of meaningful correlation. In criminal cases, this fallacy can lead to wrongful convictions when evidence is selectively framed to confirm a pre-existing hypothesis while ignoring contradictory data. In the case of Lucy Letby, did this fallacy play a significant role in shaping the prosecution’s argument?
Breaking Down the Fallacy: The “Barn Wall” of Hospital Deaths
Imagine a barn wall riddled with bullet holes.
- A skilled sharpshooter carefully aims at a pre-drawn target and hits the bullseye. This represents a genuine pattern, a case where evidence is gathered before forming a conclusion.
- A Texas sharpshooter, on the other hand, fires randomly at the barn, then paints a target around the densest cluster of bullet holes, claiming accuracy. This is a false pattern, created by selectively highlighting data that supports a conclusion while ignoring data that doesn’t.
The key mistake in the Texas Sharpshooter Fallacy is that the pattern is imposed after the data is already collected, rather than discovered through an objective analysis of all relevant information.
How This Applies to the Lucy Letby Case
1. The “Barn Wall” = All Neonatal Unit Deaths
- The neonatal unit at the Countess of Chester Hospital experienced multiple infant deaths and collapses over a specific period.
- The prosecution focused only on the subset of deaths and collapses that occurred during Letby’s shifts, effectively painting a target only after identifying her as a suspect.
- This ignores other infant deaths and medical complications that occurred during the same period when Letby was not present, much like ignoring other bullet holes on the barn wall.
2. Painting the Target Around Letby
- The prosecution used a chart showing that Letby was present at all the deaths/collapses for which she was charged.
- However, at least six other deaths during the same period were excluded from this analysis because Letby was not present for them.
- This selective focus creates a misleading illusion:
- If Letby had been present for those deaths, they likely would have been included in the charges.
- Because she was absent, they were ignored, despite potentially having the same medical causes as the deaths attributed to her.
This is a classic case of defining a pattern after seeing the data, rather than objectively analysing all neonatal deaths to determine if there was truly an unusual pattern.
Why This Statistical Error Matters
The Texas Sharpshooter Fallacy distorts the perception of probability and causation. In Letby’s case, it led to several key statistical misunderstandings:
1. Random Clustering Happens Naturally
- In any high-risk medical environment, adverse events will cluster randomly without intentional wrongdoing.
- Letby worked many shifts, increasing the likelihood that she would be present during multiple tragedies by chance alone.
- The prosecution failed to show whether other nurses, working similar hours, might also have appeared in clusters if all deaths had been analysed.
2. Base Rate Neglect: Ignoring the Expected Frequency of Nurse Presence
- The prosecution claimed that Letby’s presence at so many incidents was statistically improbable.
- But how often were other nurses present for multiple collapses?
- If most nurses worked 40% of shifts, but Letby worked 60%, she would naturally be present for more deaths.
- Without comparing her shift pattern to other nurses, the statistical claim that her presence was “too unlikely to be coincidence” is unsubstantiated.
3. Confirmation Bias: Interpreting Evidence Through a Guilt-Focused Lens
- Once Letby was identified as a suspect, investigators re-examined medical cases only from shifts she worked, looking for signs of wrongdoing.
- This ignores cases with similar medical outcomes that occurred when she was not present.
- If the same unexplained symptoms or medical complications were found in cases where Letby wasn’t working, the argument that she deliberately caused harm would weaken significantly.
4. The Prosecutor’s Fallacy: Misinterpreting Probability
- The jury was told that the probability of Letby being present for all these deaths by chance was “1 in 3.5 million”.
- This misleading argument makes two major mistakes:
- It assumes each death is an independent random event, when clusters happen naturally due to factors like seasonal infections, staffing levels, and equipment failures.
- It ignores alternative explanations, including poor hospital conditions and misdiagnosed medical complications, which might have been responsible for many of the deaths.
Expert Criticism: The Fallacy in Action
Several statisticians and medical experts have questioned the statistical reasoning behind Letby’s conviction:
- Dr. Richard Gill (Former Chair of Mathematical Statistics, Leiden University): Argued that the prosecution’s statistical argument was a “classic Texas Sharpshooter” mistake, cherry-picking data and excluding deaths where Letby wasn’t present.
- Prof. Jane Hutton (Professor of Statistics, Warwick University): Emphasised that all neonatal deaths should be analysed, not just a subset supporting the prosecution’s narrative.
- Medical Experts: Pointed out that the hospital’s mortality rate remained high even after Letby was removed from duty, suggesting systemic failures rather than the actions of a single nurse.
The Danger of the Texas Sharpshooter Fallacy in Criminal Justice
The Letby case is a textbook example of why cherry-picked statistics can create false narratives in the courtroom.
- Humans instinctively seek patterns, even in random data. When jurors see a chart where Letby’s name is the only one with multiple deaths, they may assume intent, even if the pattern is artificially constructed.
- In ambiguous medical cases, statistical manipulation can override weak physical evidence and lead to wrongful convictions.
- By focusing on Letby as a “bad actor”, the hospital avoids scrutiny over systemic failures in neonatal care, including understaffing, medical errors, and resource shortages.
The Bigger Picture: Does This Prove Letby’s Innocence?
The Texas Sharpshooter Fallacy does not prove Letby is innocent, but it does cast significant doubt on the prosecution’s statistical reasoning. When combined with:
- Disputed medical evidence (e.g. air embolism diagnoses contradicted by experts).
- No direct witnesses to wrongdoing.
- A struggling hospital with a high infant mortality rate, even after Letby’s departure.
…it suggests that the “pattern” of Letby’s presence at deaths may have been artificially constructed rather than genuinely significant.
In Justice, as in Statistics, Correlation ≠ Causation
If the jury was swayed by a pattern that was painted after the fact, then Letby may have been convicted not on solid proof, but on a fallacy. This case serves as a cautionary tale: when statistics are weaponised in courtrooms, they must be scrutinised rigorously, because mistaking correlation for causation can cost an innocent person their life.
Bayes and the Taxi Problem
A Version of this article appears in Twisted Logic: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.
THE BASICS OF THE TAXI PROBLEM
Let’s set the stage for our story. We’re in New Brighton, a city with a fleet of 1,000 taxis. Of these, 850 are blue and 150 are green. One day, a taxi is involved in an accident with a pedestrian and leaves the scene. We don’t know the colour of the taxi, and we don’t have any reason to believe that blue or green taxis are more likely to be involved in such incidents.
An independent eyewitness now comes forward. She saw the accident and claims that the taxi was green. To verify the reliability of her account, investigators conduct a series of observation tests designed to recreate the conditions of the incident. These tests reveal that she is correct about the colour of a taxi in similar conditions 80% of the time.
So, what is the likelihood that the taxi involved was actually green?
INITIAL PROBABILITIES AND INTUITIVE ESTIMATES
Your first instinct might be to believe that the chance that the taxi was green is around 80%. This assumption is based on the witness’s track record of identifying the colour of a taxi accurately. However, this conclusion doesn’t consider other crucial information—the overall number of blue and green taxis in the city.
Given the total taxi population, only 15% of them are green (150 out of 1,000), while a substantial 85% are blue. Ignoring this ‘base rate’ of taxi colours leads to a common mistake known as the ‘Base Rate Fallacy’.
APPLYING BAYES’ THEOREM TO THE TAXI PROBLEM
Bayes’ theorem is a method that helps us adjust our initial estimates based on new evidence but allowing for this base rate of the total numbers of blue and green taxis. In this way, it offers a means of updating our initial estimates after taking account of some new evidence.
For our Taxi Problem, the new evidence is the witness statement. The witness says the taxi was green, and we know that there’s an 80% chance that she is correct if the taxi was indeed green (based on her observation test). But there’s also a 20% chance that she would mistakenly say the taxi was green if it were blue.
Bayes’ theorem helps us adjust initial beliefs with new evidence, considering the base rate. Here’s how it works in the Taxi Problem:
Prior Probability: Initially, there’s only a 15% chance (150 out of 1,000 taxis) that the taxi is green.
Conditional Probability of Green Taxi (If Witness Correct): The eyewitness is correct 80% of the time.
Conditional Probability of Green Taxi (If Witness Incorrect): There’s a 20% chance the eyewitness would mistakenly identify a blue taxi as green.
After applying Bayes’ theorem, the adjusted (or ‘posterior’) probability that the taxi is green is just 41%, using the formula: ab/[ab + c (1 − a)].
THE ROLE OF NEW EVIDENCE AND MULTIPLE WITNESSES
What happens if another eyewitness comes forward? Suppose this second witness also reports that the taxi was green and, after a similar set of tests, is found to be correct 90% of the time. Now we should recalculate the probabilities using the same principles of Bayes’ theorem but including the new evidence.
The updated ‘prior’ probability is no longer the original 15%, but the 41% we calculated after hearing from the first witness. After running the numbers again, using Bayes’ formula, the revised probability that the taxi was green increases to 86%.
INTERPRETING WITNESS TESTIMONIES WITH BAYES’ THEOREM
Let’s dive a bit deeper into the implications of these results. Here are some situations that may seem counterintuitive at first, but make sense when we apply Bayes’ theorem:
The 50-50 Witness: Suppose we have a witness who is only right half the time—in other words, they are as likely to be right as they are to be wrong. Our intuition tells us that such a witness is adding no useful information, and Bayes’ theorem agrees. The testimony of such a witness doesn’t change our prior estimate.
The Perfect Witness: Now, imagine a witness who is always right—they have a 100% accuracy rate in identifying the taxi colour. In this case, if they say the taxi was green, then it must have been green. Bayes’ theorem concurs with this conclusion.
The Always-Wrong Witness: What about a witness who always gets the colour wrong? In this case, if they say the taxi is green, then it must have been blue. Bayes’ theorem agrees. We can trust this witness by assuming the opposite of what they say is the truth.
In summary, a 50% accurate witness adds no value to our estimate. A 100% accurate witness’s testimony is definitive. An always-wrong witness inversely confirms the truth.
THE BASE RATE FALLACY AND ITS IMPLICATIONS
The Base Rate Fallacy occurs when we don’t give enough weight to ‘base rate’ information (like the overall number of blue and green taxis) when making probability judgments. This mistake can lead us to overvalue specific evidence (like a single eyewitness account) and undervalue more general information like the ratio of blue to green taxis. Even so, the eyewitness may still be correct.
Again, if someone loves talking about books, we might intuitively guess that they are more likely to work in a bookstore or library than as, say, a nurse. But there are many more nurses than there are librarians or bookstore employees, and many of them love books. So, taking account of the base rate, we may well conclude that it’s more likely that the book enthusiast is a nurse than a bookstore employee or librarian.
AVOIDING THE BASE RATE FALLACY
The Base Rate Fallacy leads us to ignore general information (like the ratio of blue to green taxis or nurses to librarians) in favour of specific evidence (an eyewitness account or specific bit of information). It’s essential to balance specific and general information to avoid skewed judgments.
THE UNVEILING OF THE TRUTH
In the case of the New Brighton Taxi Problem, the mystery was solved when CCTV footage surfaced. The taxi involved was revealed to be yellow, a twist no one expected. Not really—there are no yellow taxis in New Brighton. In fact, both eyewitnesses were correct and the taxi was green.
CONCLUSION: TRUTH AND TESTIMONY
While our story was hypothetical, the principles it illustrates are very real and applicable in a wide variety of situations and circumstances. Bayes’ theorem, base rates, and new evidence are all important parts of the detective’s toolkit.
A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions, by Leighton Vaughan Williams. Published by Chapman & Hall/CRC press. 2024.
When Should We Update Our Beliefs?
Imagine emerging from a cave for the first time and watching the sun rise. You have never witnessed this before, and in this thought experiment, you are unable to tell whether it’s a regular occurrence, an infrequent event, or a once-in-a-lifetime happening.
As each day passes, however, you observe the dawn again and again: you gradually grow to expect it. With each sunrise, you become more confident that this is a regular event. With this growing confidence, you forecast that the sun will rise again the next day.
This is an illustration of what so-called Bayesian reasoning is about. Bayes’ theorem is a tool that allows us to adjust our understanding of the world based on our observations over time. It represents a process of continuous learning and understanding, pushing us gradually nearer to the truth as we are exposed to more experiences and to more information.
That’s the essence of Bayesian reasoning: adjusting our beliefs based on new information.
THE BIRTH OF BAYESIAN THINKING
The Bayesian perspective on the world can be traced to the Reverend Thomas Bayes, an 18th-century clergyman, statistician, and philosopher. The Bayesian approach advocated predicting future events based on past experiences. His ideas were in a fundamental sense different from the prevailing philosophical ideas of his time, notably those of Enlightenment philosopher David Hume.
Hume argued that we should not justify our expectations about the future based on our experiences of the past, because there is no law stating that the future will always mirror the past. As such, we can never be certain about our knowledge derived from experience. For Hume, therefore, the fact that the sun had risen every day up to now was no guarantee that it would rise again tomorrow. In contrast, Bayes provided a tool for predicting the likelihood of such events based on past experiences and observations. His method can be applied consistently to the sciences, social sciences, and many aspects of our everyday lives.
Unlike the philosopher David Hume, who argued that past experiences don’t guarantee future outcomes, Bayes focused on how we can use past events to predict the likelihood of future ones. Bayes’ approach is not just academic; it’s a practical tool.
BAYES’ THEOREM: AN EVERYDAY TOOL FOR REFINING PREDICTIONS
So how does what is known as Bayes’ theorem help us in our everyday lives and beyond? As it turns out, it’s an important way of helping us to refine our belief of what is true and what is false. Let’s look more closely into this by breaking Bayes’ theorem down into its key components:
Establish a Prior Hypothesis: The starting point in Bayesian reasoning involves the establishment of an initial hypothesis, which may or may not be true. This hypothesis, also known as the ‘prior’ belief or ‘prior probability’ that you assign to this belief being true, is based on the information available to you. For instance, if you’re trying to predict whether it will rain tomorrow, you might estimate the initial likelihood (or ‘prior probability’) based on your personal observation of current weather patterns or conditions.
Observe New Evidence: Once you establish a prior probability, you’ll then need to consider updating this when any new information becomes available. In the weather example, evidence could be anything from new dark clouds gathering or else dispersing to a sudden rise or drop in temperature.
Assess to What Extent This New Evidence Is Consistent with Your Initial Hypothesis: Bayesian reasoning doesn’t stop at just gathering evidence. It also involves considering evidence that is consistent with, or inconsistent with, your initial hypothesis. For example, if there is an increase or decrease in wind speed, this might be considered additional evidence that you should take into account in estimating the probability of rain.
Let’s break down again how Bayes’ theorem helps us refine our beliefs:
Establishing a Starting Point (The Prior Hypothesis): Imagine you’re trying to predict if it will rain tomorrow. Your ‘prior hypothesis’ is your initial estimate, based on what you currently know about the weather conditions.
Incorporating New Information (New Evidence): Now, suppose you observe unexpected dark clouds gathering in the sky. This new information should logically influence your prediction about the weather.
Combining Old and New Insights (Assessing Consistency): Bayesian reasoning involves integrating the new evidence with your initial estimate. You assess whether the appearance of dark clouds increases the likelihood of rain tomorrow.
By applying Bayes’ theorem, you adjust your belief based on the new evidence. If dark clouds often lead to rain, you increase your belief that it will rain. If not, you adjust accordingly.
Visualising Bayes’ Theorem
Think of Bayes’ theorem as a formula that combines your initial estimate with new information to give you a better estimate.
Beyond Weather: The Broad Applications of Bayes’ Theorem
Bayesian reasoning isn’t just about predicting the weather. It’s used in medicine to interpret test results, in finance to assess investment risks, in sports for game strategies, and so on. It’s a tool that refines our understanding, helping us make more informed decisions.
HOW BAYES’ THEOREM ALLOWS US TO UPDATE OUR BELIEFS
In essence, Bayes’ theorem permits us to establish an initial hypothesis, and to enter any supportive and contradicting evidence into a formula which can be used to update our belief in the likelihood that the hypothesis is true.
Consider a scenario where we evaluate our initial hypothesis. For simplicity, we label the probability that this hypothesis is correct as ‘a’. This probability is our starting point, reflecting our initial estimate based on prior knowledge or assumptions before encountering new data.
Next, we introduce ‘b’, which represents the likelihood that some new evidence we come across is consistent with our initial hypothesis being true. This is a critical element of Bayesian updating.
Conversely, ‘c’ is used to denote the probability of observing the same new evidence but under the condition that our initial hypothesis is false. This estimate is equally essential because it helps us understand the significance of the evidence in the context of our hypothesis not being true.
With these definitions in place, Bayes’ Theorem provides a powerful formula: Revised (posterior) probability that our initial hypothesis is correct = ab/[ab + c(1-a)]
This formula is a mathematical tool that updates our initial belief ‘a’ in light of the new evidence.
The result is an updated (or ‘posterior’) probability that reflects a more informed stance on the initial hypothesis.
This process, termed Bayesian updating, is a methodical approach that enables us to refine our beliefs incrementally. As we gather more evidence, we iteratively apply this updating process, allowing our beliefs to evolve closer to reality with each new piece of information. This ongoing refinement is a cornerstone of the Bayesian approach, emphasising the importance of evidence in shaping our understanding and beliefs.
BAYES’ THEOREM: A POWERFUL TOOL
Bayes’ theorem offers us a weapon against biases in our intuition, which can often mislead us. For example, intuition can sometimes lead us to ignore previous evidence or to place too much weight on the most recent piece of information. Bayes’ theorem offers a roadmap that assists us in balancing the weight of previous and new evidence correctly. In this way, it provides a method for us to fine-tune our beliefs, leading us gradually closer to the truth as we gather and consider each new piece of evidence.
CONCLUSION: THE BAYESIAN BEACON
Bayes’ theorem is more than a mathematical concept; it’s a guide through the uncertain journey of life. It teaches us to be open to new information and to continually adjust our beliefs. From daily decisions like weather predictions to complex scientific theories, Bayes’ theorem is a bridge from uncertainty to better understanding, helping us navigate life’s puzzles with more confidence and precision.
It does so in a structured way, dealing with new evidence, guiding us gradually to more informed beliefs. It encourages us always to be open to new evidence and to adjust our beliefs and expectations accordingly. Bayes’ theorem is in this sense a master key to understanding the world around us.
A Brief Look at the Hard Problem of Consciousness.
Why Does Anything Matter? A Brief Look at the Hard Problem of Consciousness
Consciousness is unlike anything else in the universe. It isn’t just about atoms, stars, or biology. It’s about what it feels like to see a sunset, to taste chocolate, or to wonder if you’ll succeed at something. Science is great at describing physical processes—how neurons fire or how eyes detect colour. But it struggles to explain why we experience anything at all. This gap between physical processes and the reality of subjective experience is often called the “hard problem of consciousness”. The “hard problem” is about more than just explaining brain activity—it’s about explaining why that activity feels like something from the inside.
What Is Consciousness?
When we talk about consciousness, we mean the “inner world” of experience:
- Seeing the colour red and knowing what red looks like (not just the wavelength data).
- Feeling sadness or joy from the inside (not just observing a brain state).
- Reflecting on your own thoughts and realising you exist.
These first-person experiences are called qualia. They’re the “what it’s like” aspect of being a thinking, feeling being. Physics and chemistry describe things from an outside perspective, like measuring weight or temperature. But how do we explain the inside perspective?
Why It’s So Surprising
Imagine an advanced alien scientist who knows every physical law and all about brain chemistry. From the outside, it can describe electrical signals and chemical exchanges. Yet nothing in those equations would capture what it’s like for you to taste chocolate or fall in love.
Under a purely naturalistic view, where everything is just matter and energy moving according to blind laws, why should anything “feel” like something from the inside? Couldn’t the universe run perfectly well with robots that respond to damage but never actually feel pain?
The Challenge for Atheism (Naturalism)
- A “Clockwork Universe”
If the universe is governed only by impersonal laws and random events, there’s no obvious reason to expect something so personal, valuable, and rich as consciousness to arise. It would be as if the universe went out of its way to create creatures who can experience beauty, love, and purpose—yet nothing in a purely blind process needs that to happen. - Fine-Tuning for Consciousness
Even if someone says that consciousness “emerges” when matter is arranged in a certain complex way (like the brain), that just restates the problem: Why does that arrangement produce an inner life, rather than merely a complex but unconscious process? On atheism, it seems like an odd coincidence that the universe not only supports life, but also subjective experience—something that can’t be measured from the outside.
Why Theism Expects Consciousness
Under theism, it makes perfect sense that beings would exist who can love, reflect on morality, and seek spiritual connection. Consciousness is not just a weird add-on—it’s a core part of the universe’s design.
- A Meaningful Creation
If there is a God who values goodness and love, then bringing about conscious creatures with the capacity for joy, moral reflection, and relationships aligns perfectly with divine intention. - Explaining “Psychophysical Laws” or “Conscious States”
Whether you believe consciousness is non-physical (dualism) or entirely physical (but still somehow experiential), theism provides a reason for why the world should generate conscious minds. It’s part of the plan for a universe filled with beings who can appreciate beauty, pursue truth, and forge deep connections.
Does the Argument Work Even If You’re a Physicalist?
Yes. Even if you think consciousness is a purely physical process in the brain, it’s still puzzling why those physical processes produce a feeling on the inside. You might say it’s “just how the brain works,” but that doesn’t explain why a cold, indifferent universe ends up with the exact conditions allowing for subjective awareness. On theism, a God wanted creatures who can think, love, and choose. So it’s not shocking that brains would be fine-tuned to do more than just process data—they’d also experience.
The Core Idea
- Atheism/Naturalism: The universe doesn’t care about feelings or meaning. All that exists is matter obeying natural laws. Consciousness ends up being a baffling stroke of luck—no one can say why it arises, only that it somehow does.
- Theism: Love, knowledge, beauty, and moral agency have a purpose beyond survival. Consciousness is central to these values, so we expect that the universe would be set up to bring about conscious beings.
In essence, if you were to guess beforehand whether a blind universe would give rise to beings that can write poetry or feel sorrow, you might guess “almost certainly not”. If, on the other hand, a personal God is creating a world for meaningful interaction and moral growth, conscious life is exactly what you’d expect.
Part of a Bigger Picture
This argument doesn’t stand alone. It often goes together with:
- Fine-Tuning: The physical constants of the universe seem set “just right” for life to exist. Consciousness is an even more delicate phenomenon than simple life.
- Moral Argument: We have an apparent sense of objective right and wrong. Consciousness is the stage where we play out moral decisions.
- Intelligibility: The universe is rationally ordered, and we, as conscious minds, can understand it. Why is that?
Together, these features suggest a universe that’s not random or indifferent but shaped to allow for life, mind, and meaning.
Conclusion: Why It Matters
Consciousness is the difference between a universe that’s a silent machine and a universe where love, art, moral reflection, and personal growth actually matter to someone. If the universe had no conscious observers, there would be no joy or sorrow, no moral responsibility, no pursuit of truth—just mindless processes.
The puzzle is: does an indifferent, purposeless reality accidentally produce such deeply meaningful experiences?
In short, the fact that our thoughts and feelings exist at all—that something as intangible as experience emerges from matter—suggests that the universe might have a deeper purpose. For many, this points beyond mere physical forces and towards the purposeful shaping of reality to include those who can truly feel, know, and care.
Ultimately, consciousness isn’t just another phenomenon; it’s the lens through which we perceive everything else. If consciousness is the heart of meaning, then perhaps meaning is at the heart of reality itself.
Further reading on the Big Questions of life can be found in Twisted Logic: Puzzles, Paradoxes, and Big Questions, by Leighton Vaughan Williams.
This is an accessible version of the full article, available at: https://leightonvw.com/2025/01/25/why-does-anything-matter/
Professor Leighton Vaughan Williams 