The telephone lines serving an airline reservation office...

The telephone lines serving an airline reservation office are all busy about $60 \%$ of the time. a. If you are calling this office, what is the probability that you will complete your call on the first try? The second try? The third try? b. If you and a friend must both complete calls to this office, what is the probability that a total of four tries will be necessary for both of you to get through?

The telephone lines serving an airline reservation office are all busy about $60 \%$ of the time.
a. If you are calling this office, what is the probability that you will complete your call on the first try? The second try? The third try?
b. If you and a friend must both complete calls to this office, what is the probability that a total of four tries will be necessary for both of you to get through?

Key Concepts

Independence of Events

Independence of events means that the outcome of one trial does not affect the outcome of another. In the telephone call scenario, each call attempt is independent; the line being busy or free on one attempt does not influence subsequent attempts. This property is crucial when calculating probabilities over multiple trials or for multiple callers.

Negative Binomial Distribution

The negative binomial distribution generalizes the geometric distribution to model the number of trials required to achieve a specified number of successes. When considering two independent callers and the total number of attempts needed for both to achieve successful connections, the problem involves the sum of two independent geometric random variables, which is a scenario described by the negative binomial distribution.

Geometric Distribution

The geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. It provides the probability of getting the first success on the k-th trial, factoring in consecutive failures beforehand. This concept is applied to compute the probability of completing a call on the first try, the second try, the third try, and so on.

Bernoulli Trials

A Bernoulli trial is a random experiment with exactly two possible outcomes: success or failure. In the context of telephone call attempts, each call can be seen as a Bernoulli trial where the call goes through (success) or does not go through (failure). The probability of success remains constant across each independent trial.

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