The sample proportion p-hat is the average of n independent Bernoulli trials with success probability p. Each trial has mean p, so the average has mean p. The variance of a single trial is p(1-p); the variance of the sum of n trials is n p(1-p). Since p-hat is the sum divided by n, its variance is (n p(1-p)) / n^2 = p(1-p)/n. For large n, the Central Limit Theorem makes p-hat approximately Normal with this mean and variance. So the correct pairing is mean p and variance p(1-p)/n. The other options mix up the mean or omit the 1/n scaling, which is why they don’t fit.

When you draw a fixed number of items from a finite population that has a known number of successes, the number of successes in your sample follows a hypergeometric distribution. The reason is that each draw changes the composition of the population, so the trials are not independent. The probability of getting exactly k successes in the sample is P(X = k) = [C(K, k) C(N − K, n − k)] / C(N, n). This counts the ways to choose k successes from the K available and n − k failures from the N − K non-successes, divided by all possible ways to choose n items from N. This differs from the binomial distribution, which assumes independent trials with a constant success probability on each draw—an assumption that fails here because removing items without replacement updates those probabilities. The Poisson and geometric distributions describe other scenarios (rare events in a large population, or the number of trials until the first success in independent trials, respectively) and don’t match the fixed-size, without-replacement setup.

In a binomial distribution, the shape and the location of the center are determined by two parameters: n, the number of trials, and p, the probability of success on each trial. The mean of the distribution is np, so these two parameters together define both how the distribution looks and where its average lies. Other options mix parameters from different distributions or misstate the mean: mu and sigma belong to the normal distribution (mean is mu), lambda is the rate parameter for the Poisson distribution, and having the mean as p ignores the effect of counting up to n trials. Therefore, the pair that defines both the shape and the mean is n and p, with mean np.

The key idea is that A and its complement A^c partition the whole sample space, so their probabilities add up to