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Chapter 6: Problem 16

A telemarketer makes six phone calls per hour and is able to make a sale on 30 percent of these contacts. During the next two hours, find: a. The probability of making exactly four sales. b. The probability of making no sales. c. The probability of making exactly two sales. d. The mean number of sales in the two-hour period.

Short Answer

Expert verified
For two hours, the probability of exactly four sales is 0.2312, no sales is 0.0138, exactly two sales is 0.2312, and the mean is 3.6 sales.

Step by step solution

01

Understand the Problem

The telemarketer makes six phone calls per hour and has a success rate of 30% for each call. We need to determine the probabilities for different sales scenarios over a two-hour period.
02

Define the Model

This problem can be modeled using a binomial distribution, where:\[ n = 12 \] (number of trials, i.e., 12 phone calls over two hours) and \[ p = 0.30 \] (probability of success on each call).
03

Find Probability for Exactly Four Sales

The probability of making exactly four sales is given by the binomial probability formula: \[ P(X = 4) = \binom{12}{4} \times 0.30^4 \times (1 - 0.30)^{12 - 4} \]. Calculate this to find the probability of exactly four sales.
04

Find Probability for No Sales

To find the probability of making no sales, use the binomial probability formula for zero successes:\[ P(X = 0) = \binom{12}{0} \times 0.30^0 \times (1 - 0.30)^{12} \]. Calculate this to determine the probability.
05

Find Probability for Exactly Two Sales

To find the probability of making exactly two sales, use:\[ P(X = 2) = \binom{12}{2} \times 0.30^2 \times (1 - 0.30)^{10} \]. Calculate this probability.
06

Calculate the Mean Number of Sales

The mean number of sales over two hours can be calculated using the formula for the mean of a binomial distribution:\[ \text{Mean} = n \times p = 12 \times 0.30 \]. Compute this to find the mean number of sales.

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Key Concepts

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

Probability Calculation
The concept of "probability calculation" is at the heart of predicting outcomes in uncertain situations. By understanding probability, we can better prepare for these uncertainties. In the telemarketer example, we can calculate the probability of different sales outcomes during a specified period.
The probability of achieving exactly four sales, no sales, or exactly two sales can be determined using the Binomial Probability Formula.
  • The formula is: \( P(X = k) = \binom{n}{k} \times p^k \times (1 - p)^{n-k} \)
  • Where \(n\) is the total number of trials (phone calls, which are 12 in this case), \(k\) is the number of successful trials (sales), \(p\) is the probability of success on each trial (30% or 0.30), and \(\binom{n}{k}\) is "n choose k", a combinatorial term.
By substituting the respective values into this formula, we can find the probability for each desired sales outcome.
Mean of Binomial Distribution
The mean of a binomial distribution is a way to find the average number of successes in a series of binary (success/failure) trials. This is particularly useful for expectations over larger sets of data. In our telemarketer example, we're calculating the mean number of sales made after many phone calls.
The mean is given by the formula:
  • \( \text{Mean} = n \times p \)
  • Where \(n\) is the number of trials, and \(p\) is the probability of success.
For our telemarketer, with 12 calls over 2 hours and a 30% success rate, the calculation would be \(12 \times 0.30\), which equals 3.6. Thus, in the long run, you can expect about 3.6 successful calls.
Success Rate in Probability
Understanding the success rate in probability is crucial for predicting how frequently a desired outcome will occur. It provides a percentage indicating how likely the success is, based on previous outcomes or given data.
In our situation, the telemarketer has a success rate of 30%. This means that statistically, each phone call has a 30% chance of resulting in a sale.
  • The probability remains constant at 0.30 (or 30%) per call, regardless of previous outcomes.
  • It represents an expectation in a single trial, influencing how we forecast outcomes over multiple trials.
Thus, while each call has a quite small chance of success, repeated trials (like calling 12 times) provide a better picture of the telemarketer's sales potential.

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