Monty Hall The Time Traveler

You find yourself in a game show. Not just any game show, but THEE game show. The Monty Hall show, hosted by TV’s very own: Monty Hall.

As the Audience claps, Monty stands in front of three doors. He speaks into his hand held microphone, “behind two of these doors are goats.” The audience laughs. “Behind the other,” he continues, “is a brand new car!”

The booming applause overwhelms your senses. After he settles down the crowd, Mr. Hall finally asks the question everyone’s been waiting for. “Which door do you pick?”

You think carefully. Each door is identical in size and color. If Monty knows which door is the right one, he’s not telling. It really doesn’t matter which you choose - they’re all identical. There’s a $\frac{1}{3}$ chance you guess right. Let’s say you pick the door to your left.

“An interesting choice,” Monty smiles mischievously, “did our contestant choose correctly?” The audience squeals in suspense. “Before we choose,” he places his free hand on your shoulder, “let’s see what you would’ve gotten if you chose the middle door.”

A drum role commences from off stage. At the climax, the door opens and the audience claps in excitement. It’s a goat.

“Phew,” he sighs, “I bet you’re happy you didn’t pick that door.” The audience erupts in laughter. “Now… I have a proposition for you.”

The audience falls silent, the tension palpable. This is the moment they’ve been waiting for.

“Do you keep your door,” he gleams, “or do you… switch… to the last door?”

The audience goes wild. They scream warnings and advice. Wild and stray “switch” and “stays” are thrown from bleachers. This is what they came here for. This is the Monty Hall problem.

The Problem

Statisticians have often used The Monty Hall problem as a brain teaser. The basic premise is the same as that of the intro story. Three doors are presented to the contestant. Behind one door, there is a special prize - usually an expensive car. The other two doors are either empty or contain a funny gag gift (i.e. goats). When the contestant chooses one door, Monty will reveal one of the doors that does not contain the prize. Finally, Monty will ask the contestant to decide whether they should keep their current choice, or switch to the remaining third door. This is the crux of the problem: to switch or to not? Which choice has the best chance of getting the big prize?

We humans are not very good at comprehending probability. You might understand that there’s a $\frac{1}{3}$ chance of picking the right door out of the three, however you might also believe that switching doors is a 50-50 choice. In reality, there is a 66% chance that you will get the car if you switch. Most people have a hard time wrapping their heads around this fact. As a result, there is an demand for media and resources discussing and dissecting the Monty Hall problem:

• Reddit Posts
• Hollywood Movies
• YouTube Videos
• Research Papers

They all lead to the correct choice. You should always switch. However, I find these sources to be unsatisfactory. While some explanations do a good job at detailing the math or provide good evidence, They do not provide me with an intuitive understanding of why switching should increase the odds. Luckily for you, I believe that the answer has been staring us in the face this whole time:

Monty Hall is a time traveler!!!

Time Traveler

To help you understand why switching doors gives you a $\frac{2}{3}$ chance, let’s imagine that Monty isn’t just revealing a door—he’s offering to let you go back in time to pick two doors instead of one. It doesn’t matter which of your two doors contain the car, just that one of them has the prize. In this modified version of the problem, you would now have a $\frac{2}{3}$ chance of picking the correct door. Given that choice, it should be fairly easy to see that switching is the better decision. What I am claiming is that switching doors is functionally equivalent to picking two doors.

Now that we’ve set the stage, let’s break down the two sequences side by side to better understand them. The tables below will help visualize how the original Monty Hall problem and our ‘time traveler’ version are essentially the same at their core.

At first, these sequences might appear to be very different. However, we can modify them a bit to make the comparison easier.

When you pick two doors, the probability of one of them having a car is $\frac{2}{3}$. But, what is the probability that one of the doors has a goat? Well, either you picked the car and a goat, or you picked two goats. There is no other possibility. Thus, when picking two doors, there is a 100% chance at least one of the doors contains a goat. No one ought to be surprised then when Monty reveals that one of those doors does in fact contain a goat. Even if this makes sense, where in the original sequence are we performing time travel? Let’s shake up the sequences one more time.

Given that there is a 100% chance that you will pick at least one door with a goat, we might as well let Monty pick that door for us. All the doors are equivalent, so this is something we can do out of the goodness of our heart. It is not, however, immediately obvious why I can make this shake up in the order of events. For the time traveler sequence, we now have Monty pick and reveal the door before we’ve even made a decision. We are then choosing that door and another door for a total of 2 choices. Wouldn’t knowingly picking a door with a goat behind it change the odds?

I would argue, however, that we should not be surprised by this door having a goat. The original probability of picking two doors does not change. There is a 100% chance one of the two doors has a goat. Knowing that does not change the fact that the probability of one of the two doors having the car is still $\frac{2}{3}$.

Comparing the sequences now, you should be able to see how they line up. We still pick one door to begin with. Next, both sequences will have Monty reveal a door to be a goat. After that, Monty will offer you to switch doors. This is the sneaky time traveling that I alluded to. He’s actually asking you if you want to go back and pick two doors. The trick is that the two doors are those that you did not pick in step 1. Finally, both sequences have you opening the last of the two doors that you’ve switched to. This time, hopefully, you have that brand new car!

The Defense

You might be skeptical at this point. Time travel? Impossible! However, this is the key missing piece that makes this brain teaser easier to grasp.

Instead of thinking of whether or not to switch as a choice between two doors. Instead, you should consider Monty offering you to go back in time and pick the other two doors. Monty revealing one of those two doors is a red-herring. Because you know that at least one of those two doors will have a goat, it doesn’t really change anything. It doesn’t matter which of the two doors has the goat, just that one of them might have the car.

At the heart of the Monty Hall problem is a simple yet powerful concept: when faced with the choice of switching, you’re not just picking between two doors. You’re leveraging a higher probability that Monty, the time traveler, has been hiding this entire time.