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Chapter 6: Problem 10

Approximately \(30 \%\) of the calls to an airline reservation phone line result in a reservation being made. a. Suppose that an operator handles 10 calls. What is the probability that none of the 10 calls results in a reservation? b. What assumption did you make in order to calculate the probability in Part (a)? c. What is the probability that at least one call results in a reservation being made?

Short Answer

Expert verified
a. The probability that none of the \(10\) calls results in a reservation is \(0.70^{10}\).\nb. The assumption made here is that each call is an independent event.\nc. The probability that at least one call results in a reservation is \(1 - 0.70^{10}\).

Step by step solution

01

Calculate the probability of no successful calls

We are given that the probability of one call resulting in a reservation (successful call) is \(0.30\). Hence, the probability of a call not resulting in a reservation (unsuccessful call) is \(1 - 0.30 = 0.70\). The operator handles \(10\) calls, and we want to know the probability that all of them are unsuccessful. Since these are independent events, we simply multiply the probabilities. So, the probability that none of the \(10\) calls results in a reservation is \(0.70^{10}\).
02

Assumptions made for the calculation

For Part (a), the assumption made is that each call is an independent event. That is, the outcome of one call (whether it results in a reservation or not) does not affect the outcomes of the other calls.
03

Determine the probability of at least one successful call

The probability of at least one successful call is the complement of the event that all calls are unsuccessful. So, its probability is \(1\) minus the probability that none of the \(10\) calls results in a reservation. From Step 1, we know the latter is \(0.70^{10}\). Hence, the probability of at least one successful call is \(1 - 0.70^{10}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events
When we talk about independent events in probability theory, it means that the outcome of one event does not influence or alter the outcome of another. In the context of the exercise problem, each call made to the airline reservation phone line is an independent event. Whether or not a particular call results in a reservation does not impact the outcome of the other calls.

This assumption of independence allows us to use the multiplication rule for probabilities. So, to find the probability of none of the 10 calls resulting in a reservation, we multiply the probability of a single call not resulting in a reservation by itself for each call, which is represented as \(0.70^{10}\). This is crucial because if the events were not independent, the probability calculations would be more complex, and the given solution would not apply.
Complement Rule
The complement rule is a helpful tool in probability that says the probability of an event happening is equal to 1 minus the probability of it not happening. This simplifies calculations, especially when it is easier to find the probability of an event not happening (given event) instead of it happening.

In part (c) of the exercise, rather than directly calculating the probability of at least one call resulting in a reservation, we calculate the probability that none of the calls result in a reservation, \(0.70^{10}\), and subtract that from 1. By doing this, we easily find the probability that at least one reservation is made. The complement rule is a powerful strategy that often turns a complex probability problem into something much more manageable.
Success and Failure Probabilities
Success and failure probabilities are two sides of the same coin in probability problems. Success usually represents the outcome of interest, while failure represents the complementary outcome. In our problem, a 'success' is making a reservation on a call, with a probability of 0.30 for each call. Conversely, a 'failure' is not making a reservation, with a probability of 0.70.

These probabilities must add up to 1, so if you know the probability of success, you can easily find the probability of failure, and vice versa. When you deal with scenarios involving multiple attempts, like the 10 calls in this exercise, calculating combined probabilities usually involves considering both successes and failures to understand the overall likelihood of various outcomes. In the exercise, knowing both probabilities is critical to finding the probability of no reservations, or at least one reservation.

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Most popular questions from this chapter

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