Let's think about a variant on the Monty Hall problem. Instead of three doors, with two goats and one car, let's think about 100 doors, with 99 goats and one car. If you pick a door at random, there is only a 1/100 chance that there's a car behind it.
Not good odds, is it?
Then Monty opens 98 doors, revealing 98 goats. Should you stick or switch?
Most of the goats have been revealed, leaving one goat door and one car door. The probability of the car being behind the door that you picked first is still 1/100, but Monty has filtered out 98/99 of the goat doors. You'd likely switch in this case, wouldn't you. It feels like the probability has 'concentrated' on the door you didn't pick.
And there's a maths explanation to do with that!
It has to do with Complementary Events. Complementary events are when an event doesn't happen. For example, if the event is 'rolling a six on a dice,' the complementary event to that is 'not rolling a six on a dice.' The probability of an event plus the probability of it's complementary event always equals one, because the event either happens or it doesn't, which makes it certain. You can find the probability of an event if you know the probability of it's complementary event, using the formula P(event)=1-P(complementary event). Complementary events are useful for all sorts of reasons, often because they are sometimes easier to work out the probability of than the event itself.
We can use complementary events here. When you first make your pick, before Monty opens the 98 doors, the chance of the car being behind your door is
P(car behind your door)= 1 door/100 doors
P(car behind your door)=1/100
And using complementary events, we can work out the probability the car is not behind your door
P(car not behind your door)=1-(1/100)
P(car not behind your door)=99/100
That isn't very helpful to us now, because even though there is a 99/100 chance it isn't behind our door, there's still 99 other doors.
But then Monty opens 98 of the goat doors, filtering out most of the doors you don't want. But P(car is behind your door) is still only 1/100, and P(car is not behind your door) is still 99/100, because the two events have to add up to one. However, there's only one other door left. After Monty filtered out the other options, the remaining door you haven't picked now has a 99/100 chance of having the car behind it. You should switch!
The same thing happens in the scenario with three doors, just in a less dramatic way. The probability of picking the car door at first is 1/3, and the probability of it not being behind your door is 2/3. When Monty filters a door, the chance the car is not behind the door you first picked is still 2/3.
(C) Zoe Hills, 2016