Can you defeat Monty Hall to win a Batmobile?

 

You slipped after accidentally stepping on a banana peel and somehow fell into another dimension where people are in game shows all the time.

As you dust yourselves off and stand up, you realize you are in the 1960s version of the game show “Let’s Make a Deal.”

The host of this show, the late Monty Hall, looks at you suspiciously at first but later presents three doors in front of you and asks you to choose one. You don’t trust strangers, so you demand to know what’s happening before you make your next move.

Monty Hall patiently explains that there’s a brand new Batmobile behind one of the doors (yes, Batman is real in this dimension), and goats behind the other doors. You could own the Batmobile if you correctly guess the door behind which it was hidden.

You pull your Batsuit out of your pocket to don the mask of the world’s greatest detective (as per DC Comics) and analyze the three doors with a careful gaze. You look meticulously for any minuscule details that might give away the correct answer to this conundrum, but the organizers behind this game show were clever enough to keep the three doors identical by removing any trace of evidence regarding which door has the Batmobile.

But undeterred, you decided not to give up and go back to the statistics and probability class you were forced to take in college a few years ago.

Thanks to the one-night cramming session you had before the final exam of that course, you vaguely remember something called the ‘principle of indifference.’ You recall that it instructs you to assign equal probabilities to all the choices you have unless there is any special evidence suggesting otherwise.

So, you assume that there’s a 1/3 chance (roughly 33%) for each door to be hiding the Batmobile. You pray to the Gods of Probability and randomly pick door 1.

You firmly close your eyes and slowly open them to match the suspense of the object behind that door being revealed. But, this is not a regular and boring game show; the host Monty has other plans for you.

He decides not to open the door you picked. Instead, he opens one of the two doors you didn’t choose and reveals a goat behind it. Later, he gives you the option to switch to the other unopened door if you wish to.

Does this change your existing odds of riding a Batmobile on the roads of this other world that evening?

Since there are only two doors now, did the chances increase from 33% to 50% for each of the two remaining doors?

If that’s the case, does it make sense for you to give up on the first door you picked, which is equally likely as the other door to be hiding the coveted prize?

Somewhere in the archives of this game show is hidden the fact that most of the regular contestants stick to their original choice believing there is no compelling reason to switch. But you summon the Gods of Probability with a deep prayer, and they will tell you that all those other contestants have been looking at this wrong!

When you picked the first door based on the principle of indifference, you effectively created two groups: the door you chose and the doors you didn’t choose. While the door you chose (the first group) has a 33% probability of hiding a Batmobile, the probability that the prize is behind one of the two other doors (the second group) was 67%.

When Monty revealed a goat in one of the doors of the second group, he didn’t magically realign the probabilities from 33%-67% to 50%-50% for these groups.

The second group still has the same chance, and since there’s only one unopened door in this group, this door’s chances increased from 33% to 67%. It provides you with counterintuitive reasoning as to why you are twice as likely to win when you switch the doors instead of sticking to your original choice.

If you think this doesn’t make any sense and there’s no reason for you not to assign equal 50% probabilities to each of the two unopened doors, let me recall the principle of indifference again.

It tells you to assign equal probabilities to choices only when there’s no ‘special evidence or information’ to suggest otherwise. In this case, Monty gave additional information about the second group of doors when he revealed the goat behind one of the doors, so the principle of indifference is no longer valid.

To give you a different kind of explanation for this, consider that you are looking to hire someone to work as a data analyst in your team. Would you hire someone after only one interview, or would you rather go for a candidate who went through multiple rounds?

You have more information regarding the skill set of the candidate after multiple interviews, and with more information comes increased confidence in that candidate’s ability to succeed in that role. Similarly, when Monty eliminated one of the doors, the other door you didn’t choose survived the test of elimination, and hence has better odds of making you the only person, other than the caped crusader himself, to own a Batmobile.