The phone lines to an airline reservation system are...

The phone lines to an airline reservation system are occupied $40 \%$ of the time. Assume that the events that the lines are occupied on successive calls are independent. Assume that 10 calls are placed to the airline. (a) What is the probability that for exactly three calls, the lines are occupied? (b) What is the probability that for at least one call, the lines are not occupied? (c) What is the expected number of calls in which the lines are all occupied?

The phone lines to an airline reservation system are occupied $40 \%$ of the time. Assume that the events that the lines are occupied on successive calls are independent. Assume that 10 calls are placed to the airline.
(a) What is the probability that for exactly three calls, the lines are occupied?
(b) What is the probability that for at least one call, the lines are not occupied?
(c) What is the expected number of calls in which the lines are all occupied?

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Key Concepts

Independent Events

Independent events are those whose outcomes do not affect the outcomes of other events. In scenarios involving multiple calls or trials, the independence assumption allows for the multiplication of individual probabilities, simplifying the analysis of the combined event outcomes.

Expected Value in a Binomial Distribution

The expected value in a binomial distribution is calculated as the product of the number of trials and the probability of success on each trial. This value represents the long-run average outcome of successes and is useful for understanding the typical behavior of the process over many repetitions.

Binomial Distribution

The binomial distribution is used to model the number of successes in a fixed number of independent trials with a constant probability of success. It is applicable in situations where each trial has only two possible outcomes (such as success and failure) and helps in calculating probabilities for events like getting exactly a certain number of successes.

Complement Rule

The complement rule is a probability principle stating that the probability of an event occurring is equal to one minus the probability of the event not occurring. This rule is particularly useful in situations where calculating the probability of the complement event is easier than calculating the event directly.

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