Mr. and Mrs. Doran have a genetic history such that the probability that a child being born to them with a certain trait is $7 / 8$ . If they have five children, what is the probability that exactly three 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

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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

7 / 8

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

\newline

Answer:

Use Binomial Probability Formula: We need to use the binomial probability formula, which is $P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$ , where: $\newline$ - $P(X=k)$ is the probability of having exactly $k$ successes in $n$ trials. $\newline$ - $\binom{n}{k}$ is the number of ways to choose $k$ successes from $n$ trials, which is calculated as $\frac{n!}{k! \cdot (n-k)!}$ . $\newline$ - $p$ is the probability of success on an individual trial. $\newline$ - $(1-p)$ is the probability of failure on an individual trial. $\newline$ - $n$ is the number of trials. $\newline$ - $k$ is the number of successes. $\newline$ In this case, $P(X=k)$ $2$ (the number of children), $P(X=k)$ $3$ (the number of children with the trait), and $P(X=k)$ $4$ (the probability of a child having the trait).
Calculate $(n \choose k)$ : First, we calculate $(n \choose k)$ for $n=5$ and $k=3$ . $(5 \choose 3) = \frac{5!}{3! \cdot (5-3)!} = \frac{(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}{((3 \cdot 2 \cdot 1) \cdot (2 \cdot 1))} = \frac{(5 \cdot 4)}{(2 \cdot 1)} = 10$ .
Calculate $p^k$ : Next, we calculate $p^k$ , which is $(\frac{7}{8})^3$ . $\newline$ $(\frac{7}{8})^3 = \frac{7}{8} * \frac{7}{8} * \frac{7}{8} = \frac{343}{512}$ .
Calculate $(1-p)^{(n-k)}$ : Then, we calculate $(1-p)^{(n-k)}$ , which is $(1 - \frac{7}{8})^{(5-3)}$ . $\newline$ $(1 - \frac{7}{8})^{(5-3)} = (\frac{1}{8})^2 = \frac{1}{64}$ .
Multiply Calculated Values: Now, we multiply all the calculated values together to find the probability. $\newline$ $P(X=3) = \binom{5}{3} \times \left(\frac{7}{8}\right)^3 \times \left(\frac{1}{8}\right)^2 = 10 \times \frac{343}{512} \times \frac{1}{64}$ .
Perform Multiplication: We perform the multiplication to get the final probability. $P(X=3) = 10 \times \frac{343}{512} \times \frac{1}{64} = \frac{3430}{32768}$ .
Convert Fraction to Decimal: Finally, we convert the fraction to a decimal and round to the nearest thousandth. $\newline$ $P(X=3) \approx \frac{3430}{32768} \approx 0.1047$ when rounded to the nearest thousandth.