# The probability space

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In a game show, a contestant is given a choice of 3 curtains. Behind one curtain is a prize, but the other two curtains conceal a sign saying "Sorry, you lose!". After the contestant chooses a curtain, the host, who knows where the prize is, will always open one of the curtains (not the one chosen by the contestant) to reveal a worthless sign. The contestant then has the option to stick with their original choice or switch to the remaining closed curtain. Model this problem in detail with a probability space. What should the contestant do?

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The probability space is depicted in this solution.

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The probability space includes all events within a sample space, and the probability associated with this event. In this particular problem, a contestant must choose from 3 curtains, two of which have a worthless sign and one of which contains the prize.

A contestant first chooses one curtain, the sample space (S) is: S = {(worthless sign),(worthless sign), (prize)} - these are the possible outcomes within the space.

The contestant must then choose whether to keep their original choice or choose the remaining curtain, the sample space then becomes S = {(worthless sign), (prize)}.

In the first choice, the probability of the contestant choosing the curtain holding the prize is 1/3, that is, one out of the possible 3 outcomes will be the one in which a prize is revealed. The probability of choosing a worthless sign is 2/3, that is, 2 out of the 3 possible outcomes will reveal a worthless sign.

In the second choice, the probability ...

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