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Male children. While it is often assumed that the probabilities of having a boy or a girl are the same, the actual probability of having a boy is slightly higher at 0.51 . Suppose a couple plans to have 3 kids. (a) Use the binomial model to calculate the probability that two of them will be boys. (b) Write out all possible orderings of 3 children, 2 of whom are boys. Use these scenarios to calculate the same probability from part (a) but using the addition rule for disjoint outcomes. Confirm that your answers from parts (a) and (b) match. (c) If we wanted to calculate the probability that a couple who plans to have 8 kids will have 3 boys, briefly describe why the approach from part (b) would be more tedious than the approach from part (a).

Playing darts. Calculate the following probabilities and indicate which probability distribution model is appropriate in each case. A very good darts player can hit the bull's eye (red circle in the center of the dart board) \(65 \%\) of the time. What is the probability that he (a) hits the bullseye for the \(10^{\text {th }}\) time on the \(15^{\text {th }}\) try? (b) hits the bullseye 10 times in 15 tries? (c) hits the first bullseye on the third try?

Stenographer's typos. A very skilled court stenographer makes one typographical error (typo) per hour on average. (a) What probability distribution is most appropriate for calculating the probability of a given number of typos this stenographer makes in an hour? (b) What are the mean and the standard deviation of the number of typos this stenographer makes? (c) Would it be considered unusual if this stenographer made 4 typos in a given hour? (d) Calculate the probability that this stenographer makes at most 2 typos in a given hour.

Lost baggage. Occasionally an airline will lose a bag. Suppose a small airline has found it can reasonably model the number of bags lost each weekday using a Poisson model with a mean of 2.2 bags. (a) What is the probability that the airline will lose no bags next Monday? (b) What is the probability that the airline will lose \(0,1,\) or 2 bags on next Monday? (c) Suppose the airline expands over the course of the next 3 years, doubling the number of flights it makes, and the CEO asks you if it's reasonable for them to continue using the Poisson model with a mean of 2.2. What is an appropriate recommendation? Explain.