The student body of a large university consists of 60% female students. A random sample of 8 students is selected. What is the probability that among the students in the sample exactly two are female?

Question

The student body of a large university consists of 60% female students. A random sample of 8 students is selected. What is the probability that among the students in the sample exactly two are female?

BlackTom AI · Accepted Answer

Begin by identifying the scenario: a random sample of 8 students from a population where 60% are female means each student has an independent 0.6 probability of being female. We are asked for the probability that exactly 2 of the 8 are female, which fits the binomial probability model with n = 8 and p = 0.6.

Option 1: 0.2936. This value is much larger than the calculated probability for exactly 2 females and seems to resemble some probability mass near the mean or for a higher count of females. It does not align with the exact binomial computation for k = 2 when p = 0.6 and n = 8, which will be smaller because getting only 2 females (out of 8) is less likely when females are the majority.

Option 2: 0.0413. This is a plausible magnitude for the exact binomial probability P(X = 2) with n = 8 and p = 0.6. To verify, one would compute the binomial term: choose(8,2) × (0.6)^2 × (0.4)^6. The combinatorial factor accounts for the number of ways to place the two female positions among eight, while the probability factors account for the likelihood of those two females and the six non-females in any specific arrangement. When calculated, this yields a value around 0.0413, matching the provided option.

Option 3: 0.0896. This value is somewhat larger than the exact P(X = 2) and could correspond to a probability of a different count of females (for instance, k = 3 or k = 1) under a particular p, but it does not correspond to the exact binomial probability for exactly two successes with p = 0.6. The discrepancy suggests a misassignment of the number of successes or a slightly altered p-value.

Option 4: 0.0007. This is a very small probability, more characteristic of extreme counts (like k = 0 or k = 1) when p is not extremely small, or of an unlikely tail event under a binomial distribution with n = 8. Since k = 2 is not an extreme tail in this scenario, this value does not reflect the exact probability of exactly two females.

To recap the reasoning pattern: we interpret the situation as binomial with n = 8 trials and success probability p = 0.6 for a female, and we seek P(X = 2). The correct approach multiplies the number of ways to choose which two of the eight are female by the probability those two are female and the remaining six are not. The other options fail either by miscounting the combinatorial term, misapplying the probability factors, or misinterpreting which count of females is being asked for. All calculations hinge on the binomial formula: P(X = k) = C(n, k) p^k (1 − p)^(n − k).

The student body of a large university consists of 60% female students. A random sample of 8 students is selected. What is the probability that among the students in the sample exactly two are female?

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The student body of a large university consists of 60% female students. A random sample of 8 students is selected. What is the probability that among the students in the sample exactly two are female?

查看解析

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类似问题

位置46的问题 Forty percent of all registered voters in a national election are female. A random sample of 5 voters is selected. The probability that there are no females in the sample is _____.0.50000.77800.07780.3456

What is the probability that at least 18 out of 20 high school students will​ graduate?

更多留学生实用工具

希望你的学习变得更简单

What is the probability that at least 18 out of 20 high school students will graduate?