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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. $20\%$ 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 exactly $2$ employees work in the marketing department? $\newline$ Write your answer as a decimal rounded to the nearest thousandth. $\newline$ ____ $\newline$

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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.

20\%

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 exactly

2

employees work in the marketing department?

\newline

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

\newline

____

\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, $n = 4$ , $k = 2$ , and $p = 0.20$ .
Calculate $C(4, 2)$ : Calculate $C(4, 2)$ which is the number of ways to choose $2$ employees out of $4$ . $C(4, 2) = \frac{4!}{2! * (4 - 2)!} = \frac{4 \times 3}{2 \times 1} = 6$ .
Calculate $(0.20)^2$ : Calculate $(0.20)^2$ which is the probability that $2$ specific employees work in marketing. $(0.20)^2 = 0.04$ .
Calculate $(1 - 0.20)^{(4 - 2)}$ : Calculate $(1 - 0.20)^{(4 - 2)}$ which is the probability that the other $2$ employees do not work in marketing. $(1 - 0.20)^{(4 - 2)} = (0.80)^2 = 0.64$ .
Multiply Values Together: Multiply all the values together to find the probability: $P(X = 2) = 6 \times 0.04 \times 0.64$ . $P(X = 2) = 0.1536$ .
Round to Nearest Thousandth: Round the answer to the nearest thousandth: $P(X = 2) = 0.154$ .

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