Conditions of bimomal model:

  1. The surveys are independent.
  2. The number of surveys is fixed.
  3. Each survey outcome can be classified as a “success” (person surveyed is female) or “failure” (person surveyed is male).
  4. The probability of a success, p, is the same for each survey, 55%.

3.41 Solution:

  1. The negative binomal model is most appropriate for calculating the probability that the 4th person surveyed is the 2nd female. The scenario satisfies conditions (1), (3), (4), and the last survey is a “success”. In other words, it describes the probability of observing the 2nd “success” on the 4th trial.
  2. Use negative binomal formula for n=4, k=2: \(\binom{n-1}{k-1} p^k {(1-p)}^{n-k}\)
choose(3,1) * 0.55^2 * 0.45^2
## [1] 0.1837688
  1. Under the binomal model for n=3, k=1, there are \(\binom{3}{1}\) = 3 combinations for 1 success in 3 trials.
  2. In the negative binomial model, outcome of the last trial is fixed and probablity of success known. Therefore, we need to find out the number of ways of ordering k-1 successes in n-1 trials. In fact, the probability of observing the kth “success” on the nth trial is exactly p times the possiblity of observing k-1 successes in n-1 trials.