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A health psychologist was researching the role of choice in food consumption. In one experiment, the psychologist offered a group of children a choice of either a chocolate chip cookie or fresh fruit. Earlier results indicate that $63\%$ of children choose the chocolate chip cookie. If the earlier results are accurate, and the psychologist randomly picks $4$ children to interview after the experiment, what is the probability that exactly $2$ of the children chose the chocolate chip cookies? Write your answer as a decimal rounded to the nearest thousandth____.

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Find probabilities using the binomial distribution

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Q. A health psychologist was researching the role of choice in food consumption. In one experiment, the psychologist offered a group of children a choice of either a chocolate chip cookie or fresh fruit. Earlier results indicate that

63\%

of children choose the chocolate chip cookie. If the earlier results are accurate, and the psychologist randomly picks

4

children to interview after the experiment, what is the probability that exactly

2

of the children chose the chocolate chip cookies? Write your answer as a decimal rounded to the nearest thousandth____.

Use Binomial Probability Formula: Use the binomial probability formula: $P(X = k) = C(n, k) \cdot (p)^k \cdot (1-p)^{(n-k)}$ Here, $n = 4$ (number of children), $k = 2$ (number of children choosing cookies), and $p = 0.63$ (probability of choosing cookies).
Calculate Combination $C(4, 2)$ : Calculate $C(4, 2)$ using the combination formula $C(n, k) = \frac{n!}{k!(n - k)!}$ . $C(4, 2) = \frac{4!}{(2! \cdot (4 - 2)!)}$ $C(4, 2) = \frac{(4 \cdot 3 \cdot 2 \cdot 1)}{(2 \cdot 1 \cdot 2 \cdot 1)}$ $C(4, 2) = 6$
Calculate $(0.63)^2$ : Calculate $(0.63)^2$ for the probability of $2$ children choosing cookies. $\newline$ $(0.63)^2 = 0.63 \times 0.63$ $\newline$ $(0.63)^2 = 0.3969$
Calculate $(1 - 0.63)^{(4 - 2)}$ : Calculate $(1 - 0.63)^{(4 - 2)}$ for the probability of the other $2$ children not choosing cookies. $\newline$ $(1 - 0.63)^{(4 - 2)} = (0.37)^2$ $\newline$ $(1 - 0.63)^{(4 - 2)} = 0.37 \times 0.37$ $\newline$ $(1 - 0.63)^{(4 - 2)} = 0.1369$
Multiply Values to Find Probability: Multiply all the values together to find the probability. $\newline$ $P(X = 2) = 6 \times 0.3969 \times 0.1369$ $\newline$ $P(X = 2) = 6 \times 0.05432721$ $\newline$ $P(X = 2) = 0.32596326$

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