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 more than one of them received five stars?

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Quizplus · Accepted Answer

The probability that more than one nursing home received five stars can be found by subtracting the probability of none or only one receiving five stars from 1. Let p = 0.21 (probability of a nursing home receiving five stars). The probability of exactly 0 or 1 nursing homes receiving five stars is calculated using the binomial probability formula. The complement of these probabilities gives the probability of more than one receiving five stars.$$ P(\text{more than one}) = 1 - [P(\text{exactly 0}) + P(\text{exactly 1})] $$$$ P(\text{exactly 0}) = \binom{7}{0} p^0 (1-p)^7 $$$$ P(\text{exactly 1}) = \binom{7}{1} p^1 (1-p)^6 $$$$ P(\text{more than one}) = 1 - [(1-0.21)^7 + 7 \times 0.21 \times (1-0.21)^6] $$Calculating these gives a value that matches the option for the probability of more than one nursing home receiving five stars, which is approximately 0.4506.