Probability of Selecting a Committee with At Least One Woman: A Combinatorial Approach

Imagine a scenario where a committee of 3 members is selected from a group of 5 men and 2 women. The question at hand is: What is the probability that the committee has at least one woman?

Combinatorial Probability: An In-Depth Analysis

To solve this problem, we can use the principle of combinatorial probability, which involves understanding how many ways we can choose items from a set without regard to the order. We will use the concept of combinations, denoted by (n) choose (r), or (binom{n}{r}), which is calculated as:

[binom{n}{r} frac{n!}{r!(n - r)!}]

Step 1: Calculate the Total Number of Ways to Select the Committee

First, we need to calculate the total number of ways to select a committee of 3 members out of the 7 people. This can be done using the combination formula:

[binom{7}{3} frac{7!}{3!(7 - 3)!} frac{7 times 6 times 5}{3 times 2 times 1} 35]

So, there are 35 possible ways to select a committee of 3 members from the 7 people.

Step 2: Calculate the Number of Ways to Select a Committee with No Women (i.e., Only Men)

Next, we calculate the number of ways to select a committee of 3 members from the 5 men. This is equivalent to selecting 3 members from 5, which can be computed as follows:

[binom{5}{3} frac{5!}{3!(5 - 3)!} frac{5 times 4}{2 times 1} 10]

Therefore, there are 10 ways to form a committee with only men.

Step 3: Calculate the Number of Committees with At Least One Woman

To find the number of committees with at least one woman, we subtract the number of all-male committees from the total number of committees:

[text{Committees with at least one woman} text{Total committees} - text{All-male committees}]

[text{Committees with at least one woman} 35 - 10 25]

So, there are 25 ways to form a committee with at least one woman.

Step 4: Calculate the Probability of Selecting a Committee with At Least One Woman

Finally, we calculate the probability of selecting a committee with at least one woman:

[P(text{at least one woman}) frac{text{Committees with at least one woman}}{text{Total committees}} frac{25}{35} frac{5}{7}]

Conclusion

Thus, the probability that the committee has at least one woman is:

[boxed{frac{5}{7}}]

Additional Context

Consider another scenario where we want to find the probability that a committee is all-male. The number of possible committees is given by:

[binom{7}{3} 35]

The number of committees with no females (only males) is:

[binom{5}{3} 10]

So, the probability of selecting a committee with only men is:

[P(text{all men}) frac{10}{35} frac{2}{7}]

Consequently, the probability of selecting a committee with at least one woman is:

[P(text{at least one woman}) 1 - P(text{all men}) 1 - frac{2}{7} frac{5}{7}]

This confirms our previous result that the probability of selecting a committee with at least one woman is (frac{5}{7}).