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$19\%$ of students in a class have black hair. If $4$ students are chosen at random, what is the probability that $0$ have black hair? Write your answer as a decimal rounded to the nearest thousandth. ____

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Find probabilities using the binomial distribution

Full solution

Q.

19\%

of students in a class have black hair. If

4

students are chosen at random, what is the probability that

0

have black hair? Write your answer as a decimal rounded to the nearest thousandth. ____

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 $p = 0.19$ .
Calculate $C(4, 0)$ : Calculate $C(4, 0)$ which is $\frac{4!}{0! (4 - 0)!}$ . Since $0!$ is $1$ and $\frac{n!}{n!}$ is $1$ , $C(4, 0)$ equals $1$ .
Solve $(0.19)^0$ : Solve $(0.19)^0$ . Any number to the power of $0$ is $1$ , so $(0.19)^0$ equals $1$ .
Simplify $(1 - 0.19)^{(4 - 0)}$ : Simplify $(1 - 0.19)^{(4 - 0)}$ . This is $(0.81)^4$ .
Calculate $(0.81)^4$ : Calculate $(0.81)^4$ . This equals $0.81 \times 0.81 \times 0.81 \times 0.81$ .
Final Calculation: $0.81 \times 0.81 \times 0.81 \times 0.81$ equals $0.43046721$ .

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