 Probability

# Geometric Distribution

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

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.

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

When flipping a fair coin repeatedly, what is the probability that the first heads appears after the $6^{\text{th}}$ but before the $9^{\text{th}}$ trial?

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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