In a 1970’s game show, a contestant was given a choice of three doors of which one contained a prize. The other two doors contained gag gifts like a chicken or a donkey. After the contestant

chose an initial door, the host of the show then revealed an empty door among the two un-chosen doors, and asks the contestant if he or she would like to switch to the other un-chosen door. The
question is should the contestant switch. Does the probability of winning increase by switching to the remaining door?

Yes, the probability of winning improves from 1/3 to 2/3 by switching. See

http://www.articlesforeducators.com/dir/mathematics/probability/the_right_door.asp

http://en.wikipedia.org/wiki/Monty_Hall_problem

excerpt:

Vos Savant's solution

The solution presented by vos Savant in Parade (vos Savant 1990b) shows the three possible arrangements of one car and two goats behind three doors and the result of switching or staying after initially picking Door 1 in each case:
Door 1 Door 2 Door 3 result if switching result if staying
Car Goat Goat Goat Car
Goat Car Goat Car Goat
Goat Goat Car Car Goat

A player who stays with the initial choice wins in only one out of three of these equally likely possibilities, while a player who switches wins in two out of three. The probability of winning by staying with the initial choice is therefore 1/3, while the probability of winning by switching is 2/3.

To answer this question, we can analyze the probabilities and assess whether switching doors increases the contestant's chances of winning.

Initially, there are three doors: A, B, and C. Let's assume the prize is behind door A. The contestant chooses door B.

Option 1: The prize is behind door A.
In this case, the host has two choices: to reveal door C or door A. However, as the host wants to make the game interesting, he must always reveal an empty door. So, he reveals door C and asks the contestant if they want to switch.

If the contestant switches to door A, they win the prize. If they stick with door B, they lose. Thus, the probability of winning by switching is 1 (the prize is certainly behind door A) and the probability of winning by sticking with the initial choice is 0.

Option 2: The prize is behind door B.
In this case, the host has only one choice: to reveal door C since he cannot reveal the prize. Once again, the host asks the contestant if they want to switch.

If the contestant switches to door A, they lose. If they stick with door B, they win. So, the probability of winning by switching is 0 and the probability of winning by sticking with the initial choice is 1.

Option 3: The prize is behind door C.
Again, the host has only one choice: to reveal door A. The host asks the contestant if they want to switch.

If the contestant switches to door B, they win. If they stick with door C, they lose. Hence, the probability of winning by switching is 1 and the probability of winning by sticking with the initial choice is 0.

From the above analysis, we can see that in two out of three cases, switching doors results in winning the prize. Therefore, the probability of winning increases if the contestant switches to the remaining door.