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 probability of being born on a weekday

The probability of a student being born on a weekday is $\frac{5}{7}$, and the probability of being born on a weekend (Saturday - Sunday) is $\frac{2}{7}$.

Step2: Use binomial probability formula

The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successful trials, $p$ is the probability of success in a single - trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 7$, $k = 4$, and $p=\frac{5}{7}$.
First, calculate the combination $C(7,4)$:
$C(7,4)=\frac{7!}{4!(7 - 4)!}=\frac{7!}{4!3!}=\frac{7\times6\times5}{3\times2\times1}=35$
Then, calculate $p^{k}\times(1 - p)^{n - k}$:
$(\frac{5}{7})^{4}\times(1-\frac{5}{7})^{7 - 4}=(\frac{5}{7})^{4}\times(\frac{2}{7})^{3}=\frac{5^{4}\times2^{3}}{7^{7}}=\frac{625\times8}{823543}=\frac{5000}{823543}$
Finally, find the probability $P(X = 4)$:
$P(X = 4)=C(7,4)\times(\frac{5}{7})^{4}\times(\frac{2}{7})^{3}=35\times\frac{5000}{823543}=\frac{175000}{823543}\approx0.213$

Answer:

$\frac{175000}{823543}$