A profoundly comprehensive Monty Hall explanation
I recently stumbled upon this video from D!NG that discusses a very famous (and VERY confusing) puzzle in probabilities, known as the Monty Hall problem.
I won’t explain the Monty Hall problem here as there are many videos and literature explaining the puzzle. Here are some sources from some really cool channels I actually follow:
I always felt that I’ve never had a fulfilling explanation that would intuitively make me understand why I should always change doors. The reason? I wasn’t paying attention!
Most of the above videos are actually showing a very intuitive explanation that I’ve noticed only but recently!
In Computer Science we are always urged to think in scale. Sometimes this kind of thinking helps a lot in maths too! So let’s rephrase the Monty Hall problem like this:
You are given 1.000 doors, 1 car and 999 goats. The car is behind door 123 and of course you don’t know it. You pick at random door 749. Monty will then open 998 doors except door 123 and the one you picked. Should you switch?
It is a no-brainer to me that in the above scenario I would ALWAYS switch doors. This is because I picked door 749 completely at random and thus the probability of choosing the one with the car is 1/1.000. But now that Monty opened 998 doors and left only 1 closed, I’m VERY confident that the car is behind door 123. Of course, there is a 1 in a thousand chance that this might be a trap! But the odds favor switching.
Now, the second trick I always like to do is to “scale down” the problem. Let’s do 100 doors and see if anything changes:
You are given 100 doors, 1 car and 99 goats. The car is behind door 12 and of course you don’t know it. You pick at random door 74. Monty will then open 98 doors except door 12. Should you switch?
Hmm, I think nothing changes! My intuition tells me again that I should switch! Let’s do one “scale down” more:
You are given 4 doors, 1 car and 3 goats. The car is behind door 2 and of course you don’t know it. You pick at random door 4. Monty will then open 2 doors except door 2. Should you switch?
I think this is where my intuition kinda breaks down. Honestly, my gut-feeling tells me, it doesn’t matter if I switch… But! From all the examples above we learned that we should always switch! Given that the rules of the game didn’t change, just the scale (number of the doors), I’m more than happy to accept that I should switch!
Now let’s change the game a tiny little bit ;)
In the previous examples we are always guaranteed that Monty will open all but 1 door after we choose our initial door. This setup actually gives us a very powerful inside: If we choose a door with a goat, Monty is forced to open all doors that have goats.
But, what if:
You are given 1.000 doors, 1 car and 999 goats. The car is behind door 123 and of course you don’t know it. You pick at random door 749. BUT Monty will then open 997 doors except doors 123 and 665. Should you switch?
Now we have 2 variables we can play with! 1) The number of doors and 2) the number of doors Monty will open after our first choice.
The answer? Yes! We should definitely switch from our initial door! To which one? It doesn’t matter! The probability between the 2 doors Monty left closed is equal!
This of course would also work if Monty leaves more than 2 doors closed after our first choice. I’ll leave it as an exercise to the reader to calculate the probabilities.
