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Xavier wrote a short fiction story about a tiny world where 76%76\% of inhabitants have a gene that allows them to breathe underwater. If 55 inhabitants are randomly chosen to be tested for the gene, what is the probability that exactly 22 have the gene? Write your answer as a decimal rounded to the nearest thousandth. ____

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Q. Xavier wrote a short fiction story about a tiny world where 76%76\% of inhabitants have a gene that allows them to breathe underwater. If 55 inhabitants are randomly chosen to be tested for the gene, what is the probability that exactly 22 have the gene? 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=2k = 2, and p=0.76p = 0.76.
  2. Calculate Combination: Calculate C(5,2)C(5, 2) using the formula C(n,k)=n!k!(n−k)!C(n, k) = \frac{n!}{k!(n - k)!}. So, C(5,2)=5!2!(5−2)!=5×42×1=10C(5, 2) = \frac{5!}{2!(5 - 2)!} = \frac{5\times4}{2\times1} = 10.
  3. Calculate Probability of 22 Inhabitants: Solve (0.76)2(0.76)^2 to get the probability of 22 inhabitants having the gene. (0.76)2=0.5776(0.76)^2 = 0.5776.
  4. Calculate Probability of Remaining: Solve (1−0.76)(5−2)(1 - 0.76)^{(5 - 2)} to get the probability of the remaining not having the gene. (1−0.76)(5−2)=(0.24)3=0.013824(1 - 0.76)^{(5 - 2)} = (0.24)^3 = 0.013824.
  5. Multiply Values Together: Multiply all the values together: P(X=2)=10×0.5776×0.013824P(X = 2) = 10 \times 0.5776 \times 0.013824. P(X=2)=0.0799584P(X = 2) = 0.0799584.
  6. Round to Nearest Thousandth: Round the answer to the nearest thousandth: P(X=2)=0.080P(X = 2) = 0.080.

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