Wonderland 2022

3. The Hallway

3. “The Hallway” - In a hallway filled with doors leading anywhere, pick one and tell us what you discover behind it. (<1000 words).

It’s like the game show problem. Pick a door any door and you could win a car! Says the game show host with such enthusiasm. He opens the first of three doors. Nothing behind that door. You now have to pick the correct door with the car behind it. Pick door one or two. If you pick door two, and there is no car, your chances are now two out of three to choose correctly.

It seems that is an old probability problem. Bertrand's box paradox, posed by Joseph Bertrand in 1889 in his Calcul des probabilités. In this puzzle, there are three boxes: a box containing two gold coins, a box with two silver coins, and a box with one of each. After choosing a box at random and withdrawing one coin at random that happens to be a gold coin, the question is what is the probability that the other coin is gold.

As in the car behind the door problem, the intuitive answer is 1 out of 2. But the probability is actually 2 out of 3. It’s called conditional probability. There is a formula for it.

Consider the event ''Ci'', indicating that the [[car]] is behind door number ''i'', takes value ''Xi'', for the choosing of the player, and value ''Hi'', for the host opening the door. The player initially chooses door i = 1, C = X1 and the host opens door i = 3, C = H3.

In this case, we have:

P(H3|C1,X1)= 1/2
P(H3|C2,X1)= 1
P(H3|C3,X1)= 0
P(Ci) = 1/3
P(Ci,Xi) = P(Ci)P(Xi)
P(H3|X1) = 1/2

Huh.

I stood in the hall pondering this problem. What if the three doors had the chance of an animal behind one of the doors. And what if that animal was a bear, or a lion, or a poisonous snake? The first door I pick is a door with nothing behind it. So by the logic above, my chances are now 2 out of three in picking the correct door. I think the chance is 1 out of 2, but the mathematicians tell me I am wrong.

I say my chances are 100% of picking the correct door. Why am I so sure of that percentage? Because I am walking down that hallway and out the door at the other end which leads to the way I came into this crazy puzzle.

W/C 411