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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? See more

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?

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In general, John and Amy are right about the weather with probabilities 70% and 60%, respectively and independently. Given the conversation above, what is the probability it rains tomorrow in their home of Brilliantia?

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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.

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Zeb's coin box contains 8 fair, standard coins (heads and tails) and 1 coin which has heads on both sides. He selects a coin randomly and flips it 4 times, getting all heads. If he flips this coin again, what is the probability it will be heads?

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Two fair dice are rolled, and it is revealed that one of the numbers rolled was a 4. What is the probability that the other number rolled was a 6?

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

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