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. ____

Question

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. ____

Bytelearn · Accepted Answer

Use Binomial Probability Formula: Use the binomial probability formula: P(X = k) = C(n, k) * (p)^k * (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) = n! / (k!(n - k)!). C(4, 2) = 4! / (2! * (4 - 2)!) C(4, 2) = (4 * 3 * 2 * 1) / (2 * 1 * 2 * 1) C(4, 2) = 6Calculate (0.63)^2: Calculate (0.63)^2 for the probability of 2 children choosing cookies. (0.63)^2 = 0.63 * 0.63 (0.63)^2 = 0.3969Calculate (1 - 0.63)^(4 - 2): Calculate (1 - 0.63)^(4 - 2) for the probability of the other 2 children not choosing cookies. (1 - 0.63)^(4 - 2) = (0.37)^2 (1 - 0.63)^(4 - 2) = 0.37 * 0.37 (1 - 0.63)^(4 - 2) = 0.1369Multiply Values to Find Probability: Multiply all the values together to find the probability. P(X = 2) = 6 * 0.3969 * 0.1369 P(X = 2) = 6 * 0.05432721 P(X = 2) = 0.32596326

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