According to the Centers for Medicare & Medicaid Services, 21% of nursing homes in the United States received five stars in overall ratings on a scale of 1 to 5. A random sample of seven nursing homes was selected. What is the probability that exactly three of them received five stars?

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Quizplus · Accepted Answer

This is a binomial probability problem where n = 7 (number of trials), k = 3 (number of successes), and p = 0.21 (probability of success on a single trial). The formula for binomial probability is P(X = k) = (nCk) * p^k * (1-p)^(n-k), where nCk is the combination of n items taken k at a time. Plugging in the values: P(X = 3) = (7C3) * (0.21)^3 * (0.79)^4 ≈ 0.1263.