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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 0.89 . If they have five children, what is the probability that exactly five of their five 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 $0$ . $89$ . If they have five children, what is the probability that exactly five of their five children will have that trait? Round your answer to the nearest thousandth. $\newline$ Answer:

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Find probabilities using the binomial distribution

Full solution

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

0

.

89

. If they have five children, what is the probability that exactly five of their five children will have that trait? Round your answer to the nearest thousandth.

\newline

Answer:

Identify values for formula: Identify the values of $n$ , $k$ , and $p$ for the binomial probability formula. $n$ represents the number of trials, which is the number of children, so $n = 5$ . $k$ represents the number of successes, which is the number of children with the trait, so $k = 5$ . $p$ represents the probability of success on a single trial, which is the probability of a child having the trait, so $p = 0.89$ .
Use binomial probability formula: Use the binomial probability formula: $P(X = k) = C(n, k) \cdot (p)^k \cdot (1-p)^{(n-k)}$ . Substitute $n = 5$ , $k = 5$ , and $p = 0.89$ into the formula to calculate the probability that exactly five of the five children will have the trait. $P(X = 5) = C(5, 5) \cdot (0.89)^5 \cdot (1-0.89)^{(5-5)}$ .
Calculate $C(5, 5)$ : Calculate the value of $C(5, 5)$ . $C(5, 5) = \frac{5!}{5!(5 - 5)!} = 1$ because the factorial of zero $(0!)$ is $1$ and any number factorial divided by itself is $1$ .
Simplify probability formula: Simplify the probability formula with the calculated values. $\newline$ $P(X = 5) = 1 \times (0.89)^5 \times (1-0.89)^{(5-5)}$ . $\newline$ Since $(1-0.89)^{(5-5)}$ is $(0.11)^0$ , and any number to the power of zero is $1$ , this term simplifies to $1$ .
Calculate $(0.89)^5$ : Calculate $(0.89)^5$ . $(0.89)^5 = 0.89 \times 0.89 \times 0.89 \times 0.89 \times 0.89$ .
Perform calculation for $(0.89)^5$ : Perform the calculation for $(0.89)^5$ . $(0.89)^5 \approx 0.5584059449$ .
Multiply values for probability: Multiply all the values together to find the probability. $\newline$ $P(X = 5) = 1 \times 0.5584059449 \times 1$ . $\newline$ $P(X = 5) \approx 0.5584059449$ .
Round answer to nearest thousandth: Round the answer to the nearest thousandth. $P(X = 5) \approx 0.558$ .

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