Discrete Mathematics

# 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.

×