Introduction

When selecting a committee from a pool of candidates, the probability of having at least one girl in the committee can be determined using mathematical principles, particularly the concept of combinations and complementary probability. This article will walk through a detailed solution to the problem of selecting a committee of 3 students from 10 candidates, where 4 of them are girls, and then extend the reasoning to a different scenario with 5 girls among 10 candidates.

Total Ways to Select a Committee

When selecting a committee of 3 students from a group of 10 candidates, the total number of ways to do this can be determined using the combination formula. Combinations are used to find the number of ways to choose a subset of items from a larger set without regard to the order of selection. The formula for combinations is given by:

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

Scenario 1: 4 Girls and 6 Boys

In this scenario, we have 10 candidates total, consisting of 4 girls and 6 boys. To calculate the total number of ways to select 3 students from these 10 candidates, we use the combination formula:

(binom{10}{3} frac{10!}{3!(10-3)!} frac{10 times 9 times 8}{3 times 2 times 1} 120)

This calculation shows that there are 120 possible ways to form a committee of 3 students from the pool of 10 candidates.

Scenario 2: 5 Girls and 5 Boys

Now, let's consider a similar scenario where there are 5 girls and 5 boys. We will use the same combination formula to find the total number of ways to form a committee of 4 students from these 10 candidates:

(binom{10}{4} frac{10!}{4!(10-4)!} frac{10 times 9 times 8 times 7}{4 times 3 times 2 times 1} 210)

Here, there are 210 possible ways to form a committee of 4 students from the pool of 10 candidates.

Probability of Selecting at Least One Girl

Instead of directly calculating the number of ways to select at least one girl, it is often easier to use the complementary probability. The complementary event is the probability of the event not happening, in this case, the probability of selecting no girls.

Scenario 1: 4 Girls and 6 Boys

First, we calculate the number of ways to select 3 boys from 6 boys:

(binom{6}{3} frac{6!}{3!3!} frac{6 times 5 times 4}{3 times 2 times 1} 20)

Using the complementary approach, the number of ways to select at least one girl is:

(120 - 20 100)

Therefore, the probability of selecting at least one girl is:

(frac{100}{120} frac{5}{6})

Scenario 2: 5 Girls and 5 Boys

To find the probability that there is no girl in the committee, we select 4 boys from the 5 boys available:

(binom{5}{4} frac{5!}{4!1!} frac{5}{1} 5)

The probability of selecting no girls in the committee is:

(frac{5}{210} frac{1}{42})

The probability of selecting at least one girl is the complement of selecting no girls:

(1 - frac{1}{42} frac{41}{42})

Conclusion

In conclusion, by using the principles of combinations and complementary probability, we can determine the likelihood of selecting a committee with at least one girl. For 4 girls and 6 boys, the probability is (frac{5}{6}), and for 5 girls and 5 boys, the probability is (frac{41}{42}). These calculations provide a clear understanding of how to approach similar probability problems in selection committees.