“Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and Monty Hall, who knows what’s behind the doors, opens another door, say number 3, which has a goat. Monty Hall then says to you, “Do you want to pick door number 2?” Is it to your advantage to switch your choice of doors?”

A central aspect of this problem is that Monty Hall KNOWS where the car and goats are. When there are two doors left and he opens one of them, he ALWAYS shows a goat. It is not a random 50:50 chance, he ALWAYS eliminates a ‘bad’ door.

Now the contestant only has two doors left, so ‘common-sense’ says that he/she has a 50:50 chance of winning and there is no difference between sticking with the original door or switching – both doors must have an even chance of winning – except they don’t!

When the contestant picked the original door it had a 1 in 3 chance of winning. Monty Hall has now eliminated a goat, (he couldn’t pick your door because you had it ‘reserved’), so your door STILL has a 1 in 3 chance of winning. The only way to get better odds is to switch doors.

When I first worked this out, I thought the odds would improve from 1 in 3 to 1 in 2. Common sense says that with two doors you have a 50:50 chance of winning. However the odds of winning are actually MUCH HIGHER than that!

To understand how, switch mind gears and think of the likliehood of losing. When you start, your door has a 2 in 3 chance of losing.  When Monty Hall shows a goat, and there is only one door left (the other door being your ‘reserved’ door), the remaining door now only has a 1 in 3 chance of losing. (One goat has gone don’t forget, so the odds must be 1 goat out of 3 choices).

BUT the only way to IMPROVE your odds is to SWITCH DOORS. Your original choice still has a 2 in 3 chance of losing, but if you switch, the remaining door only has a 1 in 3 chance of losing. So you have DOUBLED your chance of winning! 

Think of a game with 100 doors. You pick your door, and you have a 1 in 100 chance of being correct. Now Monty Hall opens 98 doors, and shows you 98 goats.

There are now 2 doors left. Your original choice which was ‘reserved’, and one other door. You can see that the odds of the other door being the winner is far higher.

If you still don’t get this, I will publish some other solutions soon.

Keep those brains well-oiled!

John McNally