Final answer:
To determine the probabilities of all committee members being female or male from the Tarbell Industries board, one uses the hypergeometric distribution, which requires calculating combinations of the respective genders and dividing by the total possible combinations.
Step-by-step explanation:
The board of directors of Tarbell Industries consists of eight men and four women. A four-member search committee was randomly selected to conduct a nationwide search for the president of the new company. The problem at hand can be classified as a hypergeometric distribution problem because we are looking at the chances of a certain composition of a committee without replacement from two distinct groups.
Part A
To calculate the probability that the four members of the search committee will be women, we use the formula for the hypergeometric distribution:
- Determine the total number of ways to choose four people from the board, which is the combination of 12 (total members) taken 4 at a time (C(12,4)).
- Calculate the number of ways to choose four women from the four available (C(4,4)).
- Divide the number of favorable outcomes for four women by the total outcomes to get the probability.
P(four women) = C(4,4) / C(12,4)
Part B
The procedure is similar for finding the probability that the four members will be men:
- Calculate the number of ways to choose four men from the eight available (C(8,4)).
- Divide the number of favorable outcomes for four men by the total outcomes.
P(four men) = C(8,4) / C(12,4)
Part C
The probabilities from part A and B do not add up to 1 because they are two distinct events within the sample space. The possibility of all committee members being male or all being female is just a small part of all potential combinations of men and women that could be selected.