Question

identifying expressions for finding probability
the math club needs to choose four people for a committee to represent it at the school board meeting. the club consists of 14 members, made up of nine girls and five boys.
which counting method should you use to find the compound probability? explain.
what is the total number of possible outcomes?
what is the probability that two girls and two boys are on the committee? round your answer to two decimal places.

Explanation:

Step1: Determine counting method

Since we are choosing a committee (where the order of selection does not matter), we use combinations. The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to be chosen.

Step2: Calculate total number of possible outcomes

We want to choose \(r = 4\) people from \(n=14\) members. Using the combination formula \(C(14,4)=\frac{14!}{4!(14 - 4)!}=\frac{14\times13\times12\times11}{4\times3\times2\times1}=1001\)

Step3: Calculate number of favorable outcomes

We want to choose \(2\) girls from \(9\) girls and \(2\) boys from \(5\) boys. Using the combination formula:

• Number of ways to choose girls: \(C(9,2)=\frac{9!}{2!(9 - 2)!}=\frac{9\times8}{2\times1}=36\)
• Number of ways to choose boys: \(C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times1}=10\)
• Number of favorable outcomes: \(C(9,2)\times C(5,2)=36\times10 = 360\)

Step4: Calculate probability

Probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{360}{1001}\approx0.36\)

Answer:

The counting method is combinations. The total number of possible outcomes is \(1001\). The probability that two girls and two boys are on the committee is \(0.36\)