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Mr. and Mrs. Doran have a genetic history such that the probability that a child being born to them with a certain trait is 63%. If they have three children, what is the probability that exactly two of their three children will have that trait? Round your answer to the nearest thousandth. Answer:

Mr. and Mrs. Doran have a genetic history such that the probability that a child being born to them with a certain trait is 63% 63 \% . If they have three children, what is the probability that exactly two of their three children will have that trait? Round your answer to the nearest thousandth.\newlineAnswer:

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Q. Mr. and Mrs. Doran have a genetic history such that the probability that a child being born to them with a certain trait is 63% 63 \% . If they have three children, what is the probability that exactly two of their three children will have that trait? Round your answer to the nearest thousandth.\newlineAnswer:
  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)} \newlineIdentify the values of nn, kk, and pp. \newlinen=3n = 3 (the number of children) \newlinek=2k = 2 (the number of children with the trait) \newlinep=0.63p = 0.63 (the probability of a child having the trait)
  2. Identify Values: Substitute the values into the binomial probability formula.\newlineP(X=2)=C(3,2)×(0.63)2×(1−0.63)3−2P(X = 2) = C(3, 2) \times (0.63)^2 \times (1-0.63)^{3-2}
  3. Calculate Binomial Coefficient: Calculate the binomial coefficient C(3,2)C(3, 2).C(3,2)=3!2!(3−2)!C(3, 2) = \frac{3!}{2!(3 - 2)!}=31= \frac{3}{1}=3= 3
  4. Calculate (0.63)2(0.63)^2: Calculate (0.63)2(0.63)^2.(0.63)2=0.63×0.63=0.3969(0.63)^2 = 0.63 \times 0.63 = 0.3969
  5. Calculate (1−0.63)(3−2)(1 - 0.63)^{(3 - 2)}: Calculate (1−0.63)(3−2)(1 - 0.63)^{(3 - 2)}.(1−0.63)(3−2)=(0.37)1(1 - 0.63)^{(3 - 2)} = (0.37)^1=0.37= 0.37
  6. Multiply Values: Multiply all the values together to find the probability.\newlineP(X=2)=3×0.3969×0.37P(X = 2) = 3 \times 0.3969 \times 0.37\newline=0.440697= 0.440697
  7. Round Answer: Round the answer to the nearest thousandth. P(X=2)≈0.441P(X = 2) \approx 0.441

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