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What is the probability that the sum of two dice will be less than 5? How many throws will it take you to hit your first "bullseye" on a dart board? Use discrete probability distributions to find out!

When throwing a fair die, what is the expected value of the number of throws needed to get a \(5?\)

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Suppose that there is a lottery which awards \(4\) million dollars to \(2\) people who are chosen at random. The price of a lottery ticket is \(10\) dollars, and a total of \(2,000,000\) people participate each time. If Russell keeps on buying lottery tickets until he wins for the first time, what is the expected value of his gains in dollars?

**Note:** The answer could be positive or negative. A negative answer indicates that Russell is expected to lose money after all.

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When throwing a fair die, what is the variance of the number of throws needed to get a \(5?\)

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Molly goes hunting with her slingshot. Her average chances of successfully hitting a target 50 meters away is \(0.05.\) What is the sum of the expected value and variance of the number of shots it takes for her to hit a bird that is 50 meters away?

**Note:** Assume that the bird stands still, even after Molly's shots.

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