Discrete Mathematics
# Conditional Probability

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.

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?

In the Monty Hall Quiz show, Monty offers the successful candidate another challenge to determine what prize he/she gets. Monty shows the contestant 3 doors, behind one door there is a new car and behind the other two doors there are goats. The contestant chooses one door and then Monty opens one of the other doors which he knows contains a goat.

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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