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# Conditional Probability

Conditional probability is the art of updating probabilities based on given information. What is the probability that the sidewalk is wet? And what if we know that it rained a few hours ago?

You are on a game show in which there are 3 doors. Behind one door is a pot of gold, but the other doors contain rocks. You choose door #1.

The host of the game show pulls a lever which will **randomly** open one of the two doors you didn't choose. To your relief, door #3 was opened and turned out to have rocks!

The host offers you a decision: stick with door #1, or switch to door #2. To maximize the probability that you get the gold, what should you do?

A family has two children. Given that one of the children is a boy, and that he was born on a Tuesday, what is the probability that both children are boys?

Assume that the probability of being born on a particular day of the week is \(\frac{1}{7}\) and is independent of whether the child is a boy or a girl.

Note: You are not told which of the numbers rolled is a 4.

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