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$56\%$ of students in a class have black hair. If $5$ students are chosen at random, what is the probability that exactly $2$ 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.

56\%

of students in a class have black hair. If

5

students are chosen at random, what is the probability that exactly

2

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

Calculate Probability of Black Hair: First, we need to calculate the probability of one student having black hair, which is $56\%$ , or $0.56$ in decimal form.
Use Binomial Probability Formula: Now, we use the binomial probability formula, which is $P(X=k) = \binom{n}{k} \cdot (p^k) \cdot ((1-p)^{n-k})$ , where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial, and $\binom{n}{k}$ is the binomial coefficient.
Identify Parameters: Here, $n=5$ (number of students chosen), $k=2$ (number of students with black hair we want), and $p=0.56$ (probability of a student having black hair).
Calculate Binomial Coefficient: Calculate the binomial coefficient $\binom{5}{2}$ , which is $\frac{5!}{2! \cdot (5-2)!}$ . That's $\frac{5\cdot4}{2\cdot1} = 10$ .
Calculate $p^k$ : Now, calculate $p^k$ , which is $0.56^2$ . That's about $0.3136$ .
Calculate $(1-p)^{(n-k)}$ : Next, calculate $(1-p)^{(n-k)}$ , which is $(1-0.56)^{(5-2)}$ . That's $0.44^3$ , which is about $0.085184$ .
Multiply All Calculations: Multiply all these together: $10 \times 0.3136 \times 0.085184$ . That's about $0.2668$ .

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