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?

This is often called the "Monty Hall" problem

Here is one explanation of the solution.
http://coastaltech.com/monty.htm

This problem is known as the Monty Hall problem, named after the host of the game show "Let's Make a Deal", Monty Hall. The Monty Hall problem is a classic probability puzzle that has generated much debate and confusion over the years.

To understand whether the contestant should switch doors, let's break down the problem and analyze it step by step.

1. Initial probability: At the beginning, when the contestant chooses one door out of three, the probability of choosing the door with the prize is 1/3. Let's assume the contestant chooses door X.

2. Revealing a door: After the contestant chooses door X, the host, Monty Hall, who knows what's behind each door, opens one of the other two doors (let's say door Y) to reveal a gag gift. The important point to note here is that Monty never opens the door that the contestant initially chose (door X), and he never reveals the door with the prize either.

3. Switching or sticking: Now, Monty presents the contestant with a decision. The contestant can either stick with their original choice (door X) or switch to the remaining unopened door (door Z). The question is, does it matter if the contestant switches or not?

To determine whether the probability of winning increases by switching, we need to consider the possible scenarios:

Scenario 1: The contestant sticks with their original choice (door X). In this case, the probability of winning remains 1/3, as it was initially.

Scenario 2: The contestant switches to the remaining unopened door (door Z). We need to analyze the probability in this scenario.

- Initially, when the contestant chose door X, the probability of winning was 1/3, as mentioned before.
- After Monty opens door Y (revealing a gag gift), the chance that the remaining door Z contains the prize increases. This is because Monty intentionally chose to open a door with a gag gift, ensuring that the prize is not behind that door.
- Now, the probability of the prize being behind door Z is 2/3 because Monty's action effectively transferred the remaining probability to that door.

Therefore, by switching, the contestant increases their chances of winning to 2/3, while sticking with the original choice only maintains the probability at 1/3.

In conclusion, the contestant should switch doors if they want to maximize their chances of winning.