Probability Calculation: The Case of Selecting All Girls From a Mixed-Gender Class

In this article, we will delve into the fascinating world of probability and combinatorics by solving a real-world problem: what is the probability that all five students picked at random from a class of 8 (consisting of 2 boys and 6 girls) are girls?

Analyzing the Problem

Let's break down the problem step by step. We need to determine the probability that all five students in a randomly selected group are girls. This involves understanding combinatorics and basic probability principles.

Step 1: Identifying the Total Number of Ways to Choose the Students

The first step is to calculate the total number of ways to choose 5 students from the 8 available. This can be calculated using the combination formula:

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

Where:

n is the total number of students (8 in this case) r is the number of students to choose (5 in this case)

Substituting the values:

binom{8}{5} frac{8!}{5!(8-5)!} frac{8!}{5! cdot 3!} frac{8 times 7 times 6}{3 times 2 times 1} 56

So, there are 56 ways to choose 5 students from 8.

Step 2: Identifying the Number of Ways to Choose 5 Girls

Next, we need to determine the number of ways to choose 5 girls from the 6 available girls. This can also be calculated using the combination formula:

binom{6}{5} frac{6!}{5!(6-5)!} frac{6!}{5! cdot 1!} 6

So, there are 6 ways to choose 5 girls from 6.

Step 3: Calculating the Probability

The probability that all students in the group are girls is the ratio of the number of favorable outcomes to the total outcomes:

P(5 girls) frac{binom{6}{5}}{binom{8}{5}} frac{6}{56} frac{3}{28}

Therefore, the probability that all five students in the group are girls is:

boxed{frac{3}{28}}

Additional Analyses

It's worth noting that this problem can be approached in different ways, each providing a deeper insight into the principles of probability and combinatorics.

Alternative Method 1: Sequential Probability Approach

Another way to approach this problem is by considering the probability of each event happening sequentially:

P(1st pick is a girl) frac{5}{8}

P(2nd pick is a girl | 1st pick is a girl) frac{4}{7}

P(3rd pick is a girl | 1st and 2nd picks are girls) frac{3}{6}

Multiplying these probabilities together gives:

P(all 3 are girls) frac{5}{8} times frac{4}{7} times frac{3}{6} frac{5}{28}

Alternative Method 2: Other Combinatorial Approaches

There are other ways to approach the problem, such as considering the total number of possible groups and the number of those that consist entirely of girls:

Total possible groups 8 choose 5 56

Total possible groups of just girls 6 choose 5 6

Probability of just girls frac{6}{56} frac{3}{28}

This confirms our earlier result.

Conclusion

In summary, the probability that all five randomly selected students from a class of 8 (2 boys and 6 girls) are girls is frac{3}{28}. This article has provided multiple methods to solve the problem, illustrating the power of combinatorics and probability in solving real-world scenarios.