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Tammy hosts a trivia night at her house every Thursday. She generates questions using a trivia website. The website claims that there is a 29%29\% chance that a generated question is a history question.\newlineIf the claim is correct, and Tammy randomly generates 55 questions during the first round of trivia, what is the probability that exactly 11 will be a history question?\newlineWrite your answer as a decimal rounded to the nearest thousandth.\newline____\newline

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Full solution

Q. Tammy hosts a trivia night at her house every Thursday. She generates questions using a trivia website. The website claims that there is a 29%29\% chance that a generated question is a history question.\newlineIf the claim is correct, and Tammy randomly generates 55 questions during the first round of trivia, what is the probability that exactly 11 will be a history question?\newlineWrite your answer as a decimal rounded to the nearest thousandth.\newline____\newline
  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=1k = 1, and p=0.29p = 0.29.
  2. Calculate C(5,1)C(5, 1): Calculate C(5,1)C(5, 1) which is the number of ways to choose 11 question out of 55. C(5,1)=5!1!∗(5−1)!=5C(5, 1) = \frac{5!}{1! * (5 - 1)!} = 5.
  3. Calculate (0.29)1(0.29)^1: Calculate (0.29)1(0.29)^1 which is the probability of getting exactly 11 history question. (0.29)1=0.29(0.29)^1 = 0.29.
  4. Calculate (1−0.29)(5−1)(1 - 0.29)^{(5 - 1)}: Calculate (1−0.29)(5−1)(1 - 0.29)^{(5 - 1)} which is the probability of not getting a history question in the other 44 questions. (1−0.29)(5−1)=(0.71)4(1 - 0.29)^{(5 - 1)} = (0.71)^4.
  5. Calculate (0.71)4(0.71)^4: (0.71)4=0.71×0.71×0.71×0.71=0.25411681(0.71)^4 = 0.71 \times 0.71 \times 0.71 \times 0.71 = 0.25411681.
  6. Calculate Probability P(X=1)P(X = 1): P(X=1)=5×0.29×0.25411681P(X = 1) = 5 \times 0.29 \times 0.25411681. Multiply all the values together to find the probability.
  7. Final Calculation: P(X=1)=5×0.29×0.25411681=0.368369165P(X = 1) = 5 \times 0.29 \times 0.25411681 = 0.368369165. Round this to the nearest thousandth.
  8. Rounded Answer: Rounded answer: 0.3680.368.

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