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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?

Bayes' Theorem

         

You flip a coin and roll a die. Let \(H\) be the event you flip a heads and let \(F\) be the event that you roll a 4. What is \(P\left(H\ | \ F\right)?\)

Note: \(P\left(H\ | \ F\right)\) denotes the probability of \(H\) occurring given that \(F\) occurs.

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You randomly choose a treasure chest to open, and then randomly choose a coin from that treasure chest. If the coin you choose is gold, then what is the probability that you chose chest A?

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A disease test is advertised as being 99% accurate: if you have the disease, you will test positive 99% of the time, and if you don't have the disease, you will test negative 99% of the time. If 1% of all people have this disease and you test positive, what is the probability that you actually have the disease?

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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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In general, the probability that it rains on Saturday is 25%.

If it rains on Saturday, the probability that it rains on Sunday is 50%. If it does not rain on Saturday, the probability that it rains on Sunday is 25%.

Given that it rained on Sunday, what is the probability that it rained on Saturday?

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