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Chapter 6: Problem 49

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 result in a reservation? b. What assumption did you make 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
The answers are: a) The probability that none of the 10 calls result in a reservation is \(0.028\). b) The assumption is that the calls are independent of each other. c) The probability that at least one call results in a reservation is \(0.972\).

Step by step solution

01

Identify the binomial distribution parameters

First, we need to identify the parameters of the binomial distribution. Here, the probability of success (p) that a call results in a reservation is \(0.30\) and the number of independent trials (n) handled by the operator is \(10\). The binomial distribution formula is given by: \[ P(x) = C(n, x) p^x (1 - p) ^ {(n - x)}]\ where P(x) is the probability of x successes in n trials, C(n,x) is the binomial coefficient 'n choose x', p is the probability of success and (1 - p) is the probability of failure.
02

Calculation of probability for Part (a)

For part (a), we are asked to find the probability that none of the 10 calls result in a reservation i.e., x=0. Substitute x = 0, n = 10 and p = 0.3 in the binomial distribution formula \[P(0) = C(10,0) (0.3)^0 (1 - 0.3)^{(10 - 0)} = (1) (1) (0.7)^{10} = 0.028\] Therefore, the probability that none of the 10 calls result in a reservation is \(0.028\).
03

Identify the assumption for Part (b)

In part (b), we need to specify the assumption made for the calculation in part (a). The assumption made here is that the calls are independent of each other, which is a basic assumption in a binomial distribution.
04

Calculation of probability for Part (c)

In part (c), we are asked to find the probability that at least one call results in a reservation. 'At least one' means 'one or more'. The probability for 'at least one' can be calculated by subtracting the probability of 'none' from 1. \[P(\text{at least one}) = 1 - P(\text{none})\]Substitute the value of P(none) from step 2: \[P(\text{at least one}) = 1 - 0.028 = 0.972\] Therefore, the probability that at least one of the 10 calls results in a reservation is \(0.972\).

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