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Bernard read an article that claimed 38%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 55 classmates, what is the probability that 00 plan to go to law school? Write your answer as a decimal rounded to the nearest thousandth.____

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Q. Bernard read an article that claimed 38%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 55 classmates, what is the probability that 00 plan to go to law school? Write your answer as a decimal rounded to the nearest thousandth.____
  1. Use Binomial Probability Formula: Use the binomial probability formula: P(X=k)=C(n,k)⋅(p)k⋅(1−p)(n−k)P(X = k) = C(n, k) \cdot (p)^k \cdot (1-p)^{(n-k)}. Here, n=5n = 5, k=0k = 0, and p=0.38p = 0.38.
  2. Calculate C(5,0)C(5, 0): Calculate C(5,0)C(5, 0) which is the number of ways to choose 00 students from 55, which is 11 because there's only one way to choose nobody.
  3. Calculate (0.38)0(0.38)^0: Calculate (0.38)0(0.38)^0 which is 11, because any number to the power of 00 is 11.
  4. Calculate (1−0.38)(5−0)(1 - 0.38)^{(5 - 0)}: Calculate (1−0.38)(5−0)(1 - 0.38)^{(5 - 0)} which is (0.62)5(0.62)^5. This is the probability that each of the 55 students does not plan to go to law school.
  5. Multiply Values Together: Now, multiply all the values together: P(X=0)=1×1×(0.62)5P(X = 0) = 1 \times 1 \times (0.62)^5.
  6. Solve (0.62)5(0.62)^5: Solve (0.62)5(0.62)^5 which is 0.62×0.62×0.62×0.62×0.620.62 \times 0.62 \times 0.62 \times 0.62 \times 0.62.
  7. Calculate Final Probability: After calculating, we get (0.62)5=0.1160292962(0.62)^5 = 0.1160292962.
  8. Round to Nearest Thousandth: So, P(X=0)=1×1×0.1160292962=0.1160292962P(X = 0) = 1 \times 1 \times 0.1160292962 = 0.1160292962.
  9. Round to Nearest Thousandth: So, P(X=0)=1×1×0.1160292962=0.1160292962P(X = 0) = 1 \times 1 \times 0.1160292962 = 0.1160292962.Round the answer to the nearest thousandth: P(X=0)≈0.116P(X = 0) \approx 0.116.

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