Based on a poll, $50 \%$ of adults believe in reincarnation. Assume that 5 adults are randomly selected, and find the indicated probability. Complate 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 (Round to three decimal places as needed.)
Answer
Rounding to three decimal places, the final answer is \(\boxed{0.156}\).
Step 1 ：This problem is a binomial probability problem. We have a binomial experiment where each trial (adult selected) is independent, there are only two possible outcomes (believe in reincarnation or not), the probability of success (believe in reincarnation) is constant (0.5), and we are interested in the number of successes in a fixed number of trials (5 adults selected).
Step 2 ：The formula for binomial probability is: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\) where: \(P(X=k)\) is the probability of k successes in n trials, \(C(n, k)\) is the combination of n items taken k at a time, p is the probability of success, n is the number of trials, and k is the number of successes.
Step 3 ：In this case, n=5 (number of adults selected), p=0.5 (probability of believing in reincarnation), and k=4 (number of adults who believe in reincarnation).
Step 4 ：Substituting the values into the formula, we get: \(P(X=4) = C(5, 4) * (0.5^4) * ((1-0.5)^(5-4))\)
Step 5 ：Calculating the above expression, we get a probability of 0.15625.
Step 6 ：Rounding to three decimal places, the final answer is \(\boxed{0.156}\).
