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Bernard read an article that claimed $38\%$ of students majoring in Political Science plan to go to law school. Curious about the claim, Bernard surveyed some of his Political Science classmates about their career goals. If the article's claim is true, and Bernard surveyed $5$ classmates, what is the probability that $0$ plan to go to law school? Write your answer as a decimal rounded to the nearest thousandth.____

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Find probabilities using the binomial distribution

Full solution

Q. Bernard read an article that claimed

38\%

of students majoring in Political Science plan to go to law school. Curious about the claim, Bernard surveyed some of his Political Science classmates about their career goals. If the article's claim is true, and Bernard surveyed

5

classmates, what is the probability that

0

plan to go to law school? 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, $n = 5$ , $k = 0$ , and $p = 0.38$ .
Calculate $C(5, 0)$ : Calculate $C(5, 0)$ which is the number of ways to choose $0$ students from $5$ , which is $1$ because there's only one way to choose nobody.
Calculate $(0.38)^0$ : Calculate $(0.38)^0$ which is $1$ , because any number to the power of $0$ is $1$ .
Calculate $(1 - 0.38)^{(5 - 0)}$ : Calculate $(1 - 0.38)^{(5 - 0)}$ which is $(0.62)^5$ . This is the probability that each of the $5$ students does not plan to go to law school.
Multiply Values Together: Now, multiply all the values together: $P(X = 0) = 1 \times 1 \times (0.62)^5$ .
Solve $(0.62)^5$ : Solve $(0.62)^5$ which is $0.62 \times 0.62 \times 0.62 \times 0.62 \times 0.62$ .
Calculate Final Probability: After calculating, we get $(0.62)^5 = 0.1160292962$ .
Round to Nearest Thousandth: So, $P(X = 0) = 1 \times 1 \times 0.1160292962 = 0.1160292962$ .
Round to Nearest Thousandth: So, $P(X = 0) = 1 \times 1 \times 0.1160292962 = 0.1160292962$ .Round the answer to the nearest thousandth: $P(X = 0) \approx 0.116$ .

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