An interviewer is given a list of people she can interview....

An interviewer is given a list of people she can interview. If the interviewer needs to interview 5 people, and if each person (independently) agrees to be interviewed with probability $\frac{2}{3},$ what is the probability that her list of people will enable her to obtain her necessary number of interviews if the list consists of (a) 5 people and (b) 8 people? For part (b), what is the probability that the interviewer will speak to exactly (c) 6 people and (d) 7 people on the list?

Key Concepts

Bernoulli Trial

A Bernoulli trial is a random experiment that has exactly two possible outcomes: success or failure. Each trial is independent, and the probability of success remains constant throughout the trials. This concept is fundamental when dealing with sequences of experiments where the outcome of each trial is binary, such as whether an individual agrees to be interviewed.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials. It provides the probability of achieving a given number of successes, given the number of trials and the constant probability of success for each trial. This distribution is essential for calculating probabilities over multiple trials when outcomes are binary.

Probability Mass Function (PMF)

The probability mass function (PMF) for the binomial distribution gives the probability of obtaining exactly k successes in n independent Bernoulli trials. It is calculated using the binomial coefficient and the probabilities of success and failure raised to the appropriate powers. This formula, P(X = k) = (n choose k) p^k (1-p)^(n-k), is key to determining the probability of a specific number of successful outcomes.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for the binomial distribution provides the probability that the number of successes is less than or equal to a certain value. This concept is used when calculating the probability of achieving at least or at most a given number of successes, by summing the probabilities given by the binomial PMF over the desired range of outcomes.

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