Question

1. a class has seven students. what is the probability that exactly four of the students were born on a weekday (monday through friday)?

Explanation:

Step1: Determine the probability of being born on a weekday and weekend

There are 7 days in a week. The number of weekdays (Monday - Friday) is 5, so the probability \(p\) of a student being born on a weekday is \(p=\frac{5}{7}\). The probability \(q\) of a student being born on a weekend (Saturday - Sunday) is \(q = 1 - p=1-\frac{5}{7}=\frac{2}{7}\).

Step2: Use the binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times q^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successful trials, \(C(n,k)=\frac{n!}{k!(n - k)!}\) is the number of combinations. Here, \(n = 7\), \(k = 4\).
First, calculate \(C(7,4)=\frac{7!}{4!(7 - 4)!}=\frac{7!}{4!3!}=\frac{7\times6\times5}{3\times2\times1}=35\).
Then, \(p^{4}=(\frac{5}{7})^{4}\) and \(q^{7 - 4}=q^{3}=(\frac{2}{7})^{3}\).
So \(P(X = 4)=C(7,4)\times(\frac{5}{7})^{4}\times(\frac{2}{7})^{3}\).

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Answer:

The probability that exactly four of the seven students were born on a weekday is \(\frac{175000}{823543}\approx0.2125\)