When Should We Switch to Survive?

The Deadly Doors Dilemma

THE GAME OF DESTINY: CHOOSING BETWEEN FOUR DEADLY DOORS

Welcome to a dark game of chance and destiny, featuring four distinctive doors coloured red, yellow, blue, and green. Three of these gateways lead to an instant, dusty demise, while the remaining one offers a golden path to fame and fortune. The destiny of each door is randomly assigned by the host, who picks out four coloured balls from a bag—red, yellow, blue, and green. This random process determines the fate that each door offers.

THE INITIAL CHOICE AND ODDS OF SURVIVAL

Suppose you find yourself drawn to the red door. Given the game’s rules, your chance of picking the lucky door and moving onto a path of wealth and glory stand at just one in four, or 25%. Conversely, the unnerving possibility of your choice leading to a dusty doom looms large, with a daunting chance of three in four, or 75%. This calculation comes directly from the fact that out of the four doors, only one leads to fortune, while the other three lead to an unwelcome demise.

A TWIST IN THE TALE: THE HOST’S REVEAL

But the game involves a twist: the host, who knows where each door leads, opens one of the remaining doors. In this case, he reveals the yellow door to be one of the deadly ones. This is a part of the game’s rule—the host must open a door after the initial choice, revealing one of the deadly doors while leaving the lucky door unopened.

THE PIVOTAL DECISION: TO SWITCH OR NOT TO SWITCH

With one door opened and its deadly fate exposed, you face a critical decision. Would you stick with your original choice, the red door, or change your fate by choosing either the blue or green door? This predicament is an extension of the classic three-door Monty Hall Problem, which we can term ‘Monty Hall Plus’, but the underlying logic is exactly the same.

THE COMMON MISCONCEPTION: MISUNDERSTANDING PROBABILITIES

Intuition might suggest that with one door less in the equation, the chance of the red door leading to fortune must have improved. After all, now there are only three doors left—the red, blue, and green. If we assume each door is now equally likely to be the lucky one, the probability of each would be one in three.

ANOTHER REVEAL, ANOTHER DEATH TRAP

However, the host has yet to finish his part. He proceeds to open another door, unveiling the blue one this time, which again turns out to be a death trap. Now, with only two doors remaining—the red and green—the odds seem to have further improved, right? The likelihood of each door leading to fortune should now stand at a clear 50-50, or does it? Does it matter if you stick with your original choice or switch to the remaining door?

THE COUNTERINTUITIVE TRUTH: WHY THE INITIAL CHOICE MATTERS

Contrary to intuitive reasoning, the answer is a resounding yes; it does matter if you stick or switch. The reason for this lies in the fact that the host knows what lies behind each door. When you initially chose the red door, your odds of it leading to fame and fortune were 25%. These odds remain unchanged if you persist with your original choice, regardless of which doors the host reveals subsequently.

THE VALUE OF INFORMATION: HOW THE HOST’S ACTIONS ALTER THE ODDS

Here lies the crux of the game—the host’s actions, since they are informed, change the probabilities associated with the remaining doors. Before the host opened the yellow door, there was a 75% chance that the fortunate door was one among the yellow, blue, or green doors. But now, with the yellow door revealed as deadly, the same 75% probability now gets distributed to the remaining doors, i.e. the remaining (blue and green) doors.

THE FINAL REVEAL: GREEN—THE FINAL OPTION

As the host opens the blue door, unveiling yet another deadly fate, the odds shift again. The chance of the green door being the fortunate one grows further, given that it is now the only door standing against your initial choice, the red door. Therefore, you could either stick with your original choice and hold onto the 25% chance of survival, or switch to the green door, enhancing your odds to a favourable 75%. Essentially, the combined probability of the doors not initially chosen (which was originally 3/4) now heavily favours the last unopened door (since two of three potential safe doors have been eliminated).

CONCLUSION: THE IMPLICATION OF KNOWLEDGE

This dynamic interplay of choices and probabilities is a result of the host’s knowledge about what lies behind each door. The host’s actions introduce new information into the game and influence the probability associated with the remaining unopened doors. The odds change because the host, knowing the outcomes, will never inadvertently reveal the lucky door. However, if the host didn’t possess this knowledge and the doors were revealed randomly, the game would lose its strategic aspect and boil down to sheer luck. In such a scenario, if two doors remain, the chances would be a clear 50-50, making a coin toss as effective a decision-making tool as any others.