When Should We Roll the Dice?

A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman and Hall/CRC Press, 2024).

UNDERSTANDING THE CHEVALIER’S DICE PROBLEM

Probability is the science of uncertainty, providing a way to measure the likelihood of events occurring. It can be viewed as a measure of relative frequency or as a degree of belief. In the context of gambling, understanding probability is crucial for making informed decisions and avoiding common pitfalls.

A famous problem, known as the Chevalier’s Dice Problem, sheds light on the some of the intricacies of probability.

To understand the problem, it is essential to grasp some fundamental concepts of probability. Consider a single die roll—each outcome represents a possible event, such as rolling a 1, 2, 3, 4, 5, or 6. When rolling two dice, there are 36 possible outcomes (six outcomes for the first die multiplied by six outcomes for the second die).

THE FLAWED REASONING OF THE CHEVALIER

The Chevalier’s Dice Problem originated from a gambling challenge offered by the Chevalier de Méré, a 17th-century French gambler. The Chevalier offered even money odds that he could roll at least one six in four rolls of a fair die.

The Chevalier’s reasoning was based on the assumption that since the chance of rolling a six in a single die roll is 1/6, the probability of rolling a six in four rolls would be 4/6 or 2/3. However, this reasoning can be shown to lead to inconsistent results when extrapolated to more rolls.

The correct approach involves considering the independent nature of each throw of the die. The probability of a six in one go is 1/6, so the probability of not getting a six on that go is 5/6. To calculate the probability of not rolling a six in four throws, we multiply the probabilities: (5/6) × (5/6) × (5/6) × (5/6) = 625/1296.

Therefore, the probability of at least one six in four attempts is obtained by subtracting the probability of not rolling a six in any of those four attempts from 1: 1 − (625/1,296) = 671/1,296 ≈ 0.5177, which is greater than 0.5.

Despite his faulty reasoning, the Chevalier still had an edge in this game by offering even money odds on an event with a probability of 51.77%.

THE CHEVALIER’S MISSTEP WITH THE MODIFIED GAME

Encouraged by his initial success, the Chevalier expanded the game to 24 rolls of a pair of dice, betting on the occurrence of at least one double-six. His reasoning followed the same flawed pattern: since the chance of rolling a double-six with two dice is 1/36, he believed the probability of at least one double-six in 24 rolls would be 24/36 or 2/3.

The correct probability calculation involved considering the independent nature of each dice roll. The probability of no double-six in one roll is 35/36. Therefore, the probability of no double-six in 24 rolls is (35/36) raised to the power of 24, which is approximately 0.5086.

Subtracting this value from 1 yields the probability of at least one double-six in 24 rolls: 1 − 0.5086 = 0.4914, which is less than 0.5. Hence, the Chevalier’s edge in this modified game was negative: 49.14% − 50.86% = −1.72%.

This outcome demonstrated that even if the odds seem favourable, incorrect reasoning can lead to erroneous conclusions. The Chevalier’s faulty understanding of probability caused him to lose over time.

THE IMPORTANCE OF CORRECT PROBABILITY CALCULATION

These examples underscore the critical nature of accurate probability calculations in games of chance. While intuitive reasoning may seem convincing, it often leads to incorrect conclusions, as demonstrated by the Chevalier’s bets. Understanding the true probability of events is essential for informed decision-making in gambling and many other contexts where risk and uncertainty play significant roles.

THE GAMBLER’S RUIN AND UNDERSTANDING FINITE EDGES

The Gambler’s Ruin problem raises the complementary question of whether, in a gambling game, a player will eventually go bankrupt if playing for an extended period against an opponent with infinite funds, even if the player has an edge.

For instance, imagine a fair game where you and your opponent flip a coin, and the loser pays the winner £1. If you start with £20 and your opponent has £40, the probabilities of you and your opponent ending up with all the money can be calculated using the following formulas:

P1 = n1/(n1 + n2); P2 = n2/(n1 + n2)

Here, n1 represents the initial amount of money for player 1 (you) and n2 represents the initial amount for player 2 (your opponent). In this case, you have a 1/3 chance of winning the £60 (20/60), while your opponent has a 2/3 chance. However, even if you win this game, playing it repeatedly against various opponents or the same one with borrowed money will eventually lead to the loss of your betting bank. This holds true even when the odds are in your favour. This is an important lesson in risk management, emphasising the importance of not only the odds but also the size of one’s bankroll relative to the stake sizes.

The Gambler’s Ruin problem, as explored by Blaise Pascal, Pierre Fermat, and later mathematicians like Jacob Bernoulli, reveals the inherent risks of prolonged gambling, even with favourable odds.

PILOT ERROR: MISUNDERSTANDING CUMULATIVE PROBABILITY

In Len Deighton’s novel ‘Bomber’, a statistical claim suggests that a World War II pilot with a 2% chance of being shot down on each mission is ‘mathematically certain’ to be shot down after 50 missions. This assertion is a classic example of misinterpreting cumulative probability. In reality, if a pilot has a 98% chance of surviving each mission, their probability of not being shot down after 50 missions is 0.98 to the power of 50 (0.9850)which is approximately 0.36, or 36%. Thus, their chance of being shot down over these 50 missions is 64% (1 − 0.36), not 100%.

SURVIVORSHIP BIAS: THE CASE OF BULLET-RIDDEN PLANES

The concept of survivorship bias is vividly illustrated in the case of analysing planes returning from missions during World War II. Upon examining these planes for bullet holes, it was observed that most hits were on the wings, tail, and the body of the plane, with few on the engine. The initial, intuitive response might be to reinforce the areas with the most bullet holes. However, this would be a misinterpretation of the data.

The key realisation, identified by statistician Abraham Wald, was that the planes being analysed were those that survived and returned to base. The areas with fewer bullet holes, such as the engines, were likely critical to survival. Planes hit in these areas probably didn’t make it back, hence the lack of data for these hits. This understanding exemplifies survivorship bias—focusing on survivors (or what’s visible) can lead to incorrect conclusions about the whole population.

Wald’s insight led to the reinforcement of seemingly less-hit areas like engines, contributing significantly to the survival of many pilots. His work in operational research during the war provided a critical perspective on interpreting data and making decisions under uncertainty.

CONCLUSION: DICE, ODDS, AND RUIN

The Chevalier’s Dice Problem illustrates the importance of understanding probability in gambling scenarios. Probability theory, as developed through famed correspondence between Pascal and Fermat, has contributed to modern probability concepts and the understanding of risk involved in gambling.

The Gambler’s Ruin is a kind of warning from the world of probability, telling us that in gambling, a slight edge is no guarantee of success. Imagine two gamblers, one with an edge over the other but with much less money to play with. Even if the first player is more likely to win each round, their thinner wallet means they could run out of money after a few bad games. In contrast, the player with the deep pockets can keep playing longer, until (given enough money) luck swings their way. This underlines the importance and impact of losing streaks in games of chance.

The wartime examples highlight the real-world importance of understanding probability and statistical concepts accurately. They serve as a reminder that intuition can often lead us astray. Correctly interpreting data, especially in high-stakes situations, can have life-saving implications.