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A social media company holds weekly meetings, which all employees are required to attend. At these meetings, the head of the company randomly selects employees to share an accomplishment from the past week. $21\%$ of company employees work in the marketing department. $\newline$ If the head of the company randomly chooses $4$ employees to share accomplishments at the next meeting, what is the probability that $0$ employees work in the marketing department? $\newline$ Write your answer as a decimal rounded to the nearest thousandth. $\newline$ ____

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Find probabilities using the binomial distribution

Full solution

Q. A social media company holds weekly meetings, which all employees are required to attend. At these meetings, the head of the company randomly selects employees to share an accomplishment from the past week.

21\%

of company employees work in the marketing department.

\newline

If the head of the company randomly chooses

4

employees to share accomplishments at the next meeting, what is the probability that

0

employees work in the marketing department?

\newline

Write your answer as a decimal rounded to the nearest thousandth.

\newline

____

Use Binomial Probability Formula: Use the binomial probability formula: $P(X = k) = C(n, k) \cdot (p)^k \cdot (1-p)^{(n-k)}$ . Here, $n = 4$ , $k = 0$ , and $p = 0.21$ .
Calculate $C(4, 0)$ : Calculate $C(4, 0)$ using the formula $\frac{n!}{k!(n - k)!}$ . Since $k = 0$ , $C(4, 0) = \frac{4!}{0!(4 - 0)!} = 1$ .
Solve $(0.21)^0$ : Solve $(0.21)^0$ . Any number to the power of $0$ is $1$ , so $(0.21)^0 = 1$ .
Simplify $(1 - 0.21)^{(4 - 0)}$ : Simplify $(1 - 0.21)^{(4 - 0)}$ . Calculate $(0.79)^4 = 0.79 \times 0.79 \times 0.79 \times 0.79$ .
Calculate $P(X = 0)$ : $P(X = 0) = 1 \times 1 \times (0.79)^4$ . Calculate $(0.79)^4 = 0.389017$ .

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