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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? See more

# Level 3

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?

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?

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.

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?

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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