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 two or three of them received five stars?

Question

Quizplus · Accepted Answer

The probability of two or three nursing homes receiving five stars can be calculated using the binomial probability formula, which is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success. For two successes (k=2), the calculation is P(X = 2) = (7 choose 2) * (0.21)^2 * (0.79)^5. For three successes (k=3), it's P(X = 3) = (7 choose 3) * (0.21)^3 * (0.79)^4. Adding these two probabilities gives the total probability of having either two or three nursing homes with five stars, which is approximately 0.4112.