Based on a poll, $40 \%$ of adults believe in reincarnation. Assume that 5 adults are randomly selected, and find the indicated probability. Complete parts (a) through (d) below. a. What is the probability that exactly 4 of the selected adults believe in reincarnation? The probability that exactly 4 of the 5 adults believe in reincarnation is 0.077 . (Round to three decimal places as needed.) b. What is the probability that all of the selected adults believe in reincarnation? The probability that all of the selected adults believe in reincarnation is 0.010 . (Round to three decimal places as needed.) c. What is the probability that at least 4 of the selected adults believe in reincarnation? The probability that at least 4 of the selected adults believe in reincarnation is 0.087 . (Round to three decimal places as needed.) d. If 5 adults are randomly selected, is 4 a significantly high number who believe in reincarnation? A. Yes, because the probability that 4 or more of the selected adults beflieve in reincarnation is greater than 0.05 . B. Yes, because the probability that 4 or more of the selected adults believe in reincarnation is less than 0.05 . C. No, because the probability that 4 or more of the selected adults believe in reincarnation is less than 0.05 . D. No, because the probability that 4 or more of the selected adults believe in reincarnation is greater than 0.05 . Clear all Check answer
Solution
Step 1 ：Given that the probability of an adult believing in reincarnation is 40%, we are asked to find the probability that exactly 4 out of 5 randomly selected adults believe in reincarnation. This is a binomial probability problem, where we have \(n=5\) trials (the number of adults), \(k=4\) successes (the number of adults who believe in reincarnation), and \(p=0.4\) (the probability of success on each trial). The formula for binomial probability is: \[P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\] where \(C(n, k)\) is the binomial coefficient, which gives the number of ways to choose \(k\) successes out of \(n\) trials.
Step 2 ：Substituting the given values into the formula, we get: \[P(X=4) = C(5, 4) * (0.4^4) * ((1-0.4)^(5-4))\]
Step 3 ：Calculating the above expression, we find that the probability is approximately 0.077.
Step 4 ：Final Answer: The probability that exactly 4 out of 5 randomly selected adults believe in reincarnation is approximately \(\boxed{0.077}\).
