Exercise 29

Sampling at school. For a sociology class project you are asked to conduct a survey on 20 students at your school. You decide to stand outside of your dorm’s cafeteria and conduct the survey on a random sample of 20 students leaving the cafeteria after dinner one evening. Your dorm is comprised of 45% males and 55% females.

1. Which probability model is most appropriate for calculating the probability that the $4^{th}$ person you survey is the $2^{nd}$ female? Explain.

2. Compute the probability from part (a).

3. The three possible scenarios that lead to $4^{th}$ person you survey being the $2^{nd}$ female are

   $$\{M,M,F,F\},\{M,F,M,F\},\{F,M,M,F\}$$

   One common feature among these scenarios is that the last trial is always female. In the first three trials there are 2 males and 1 female. Use the binomial coefficient to confirm that there are 3 ways of ordering 2 males and 1 female.

4. Use the findings presented in part (c) to explain why the formula for the coefficient for the negative binomial is ${n-1\choose k-1}$ while the formula for the binomial coefficient is ${n\choose k}$.