A number of years ago, there was a popular television game show called Let's Make a Deal. The host, Monty Hall, would randomly select contestants from the audience and, as the title suggests, he would make deals for prizes. Contestants would be given relatively modest prizes and then would be offered the opportunity to risk that prize to win better ones.
Suppose you are a contestant on this show. Monty has just given you a free trip worth $500 to a locale that is of little interest to you. He now offers you a trade: Give up the trip in exchange for a gamble. On the stage are three curtains, A, B, and C. Behind one of them is a brand-new car worth $45,000. Behind the other two curtains, the stage is empty.
You decide to gamble and give up the trip. (The trip is no longer an option for you.) You must now select one of the curtains. Suppose you select Curtain A.
In an attempt to make things more interesting, Monty then exposes an empty stage by opening Curtain C (he knows that there is nothing behind Curtain C). He then asks you if you want to keep Curtain A, or switch to Curtain B.
What would you do?
Hint: Questions to consider are: What is the probability of winning and the probability of losing the car prior to opening Curtain C? What is the probability of winning and the probability of losing the car after Curtain C is opened? What is your best strategy?
This is the classic Monty Hall problem and the answer is that you will increase your chances by switching your choice.
If you don't - you have not changed your probability to win - it is still 1/3
However, once Monty opens a door with nothing behind, it changes the information you have in a beneficial way - you always eliminate an ...
The solution discusses the probability of winning by switching to a different curtain. It discusses the Monty Hall problem in detail with different configurations. A reference link is provided.