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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?

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

1% of people have a rare cancer, and there is a test for this cancer which is "90% accurate"; that is:

If you have the cancer there is a 90% chance the test will be positive.

If you don't have the cancer there is a 90% chance the test will be negative.

If you take the test and test positive, what is the approximate probability that you have the cancer?

He then offers the contestant the chance of changing his choice. Should the contestant switch to the other unopened door?

A. No, he shouldn't because he should trust his first instinct.

B. No, he shouldn't because Monty is trying to trick him.

C. Yes, he should because the odds have changed.

D. Maybe, but his odds haven't changed.

10% of all items produced are defective. The worker who chooses items for inspection has a sense for which are defective, so 60% of all defective items are inspected, while only 20% of all good items are inspected. If an item is inspected, what is the probability it is defective?

Enter your answer as a decimal.

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