Probability of Committee with Six Women

A certain union chapter has 32 male and 26 female members, totaling 58 members. If a committee of 7 people is randomly selected, what is the probability that the committee has exactly 6 women?

This problem requires an understanding of discrete probability within a combinatorial context. To solve this, one must first identify the total number of possible committee compositions that can be formed from the given set of union members. This involves using combinations, where the number of ways to choose a specified number of individuals from a larger set is calculated. The specific task here is to calculate the probability of forming a committee with exactly six women out of the total of seven people.

In such probability questions, it is key to separately calculate the number of favorable outcomes (in this case, choosing six women and one man) from the total number of possible outcomes (choosing any seven people from fifty-eight). The concept of combinations is essential as we leverage combinatorial selection techniques: choosing six from twenty-six women and, simultaneously, one from thirty-two men.

Once the favorable sets and total sets are determined, the probability is computed as the ratio of favorable outcomes to the total possible outcomes. This problem introduces the fundamental concept of combining probability theory with combinatorial principles, serving as a foundation for more complex stochastic problems in discrete mathematics. It integrates the understanding of how discrete structures are used to model probabilities, a core skill in both theoretical and applied contexts of discrete math.