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?

Key Concepts

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.

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.

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.

Transcript

00:01 When placing a phone call to an airline reservation system, there's a 40 % chance that the phone lines will all be occupied that our phone call will not go through.

00:17 So the probability that the phone lines were all occupied when placing a call is 40 % or .4.

00:27 So p equals .4.

00:29 We are going to place 10 calls.

00:38 So there are going to be 10 trials.

00:43 Now when we place these 10 calls, what is the probability that for exactly three of these 10 calls, the lines were all occupied? so we're calling the airline reservation system 10 times.

01:03 What's the probability that out of those 10 calls that we place, that exactly three of them don't go through because the lines are all occupied.

01:15 Well, we're working with a binomial random variable.

01:20 X represents the number of successes.

01:27 In this case, we're calling a success when the lines are all occupied.

01:34 So what's the probability that x equals three? out of 10 calls, out of 10 trials, was the probability of having three successes that the lines were occupied exactly three times out of our 10 calls.

01:58 So what's the probability that x equals 3? well, and the total number of phone calls that we are placing is 10.

02:06 The probability of success, the probability of success in this case means the lines are all occupied.

02:15 So the probability of success, we mean the probability that the lines were all occupied.

02:21 That's 0 .4.

02:22 Lines are all occupied 40 % of the time.

02:25 So the probability that the line is occupied, 0 .4.

02:30 So there's a graph of our binomial distribution.

02:33 Now, out of the 10 calls was the probability.

02:38 That the lines were all occupied exactly three times.

02:45 Probability of three successes, that three out of our 10 calls, exactly three out of our 10 calls resulted in the call not being able to go through because the lines were all occupied.

02:57 Probability that x equals 3 .215.

03:07 All right...

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