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Chapter 4: Problem 106

A heavy-equipment salesman can contact either one or two customers per day with probabilities \(1 / 3\) and \(2 / 3,\) respectively. Each contact will result in either no sale or a \(\$ 50,000\) sale with probabilities \(9 / 10\) and \(1 / 10,\) respectively. What is the expected value of his daily sales?

Short Answer

Expert verified
Answer: The expected value of daily sales is approximately $10,667.

Step by step solution

01

Identify possible outcomes

First, we will identify the possible outcomes and their corresponding probabilities. There are three possible outcomes: no sale (\(0), one sale (\)50,000), and two sales ($100,000).
02

Calculate probabilities of each outcome

Next, we need to calculate the probability of each outcome. We will do this using the probabilities provided for contacting customers and making sales. Probability of no sale: $$\frac{1}{3}\times \frac{9}{10} + \frac{2}{3}\times \frac{9}{10}\times \frac{9}{10} = \frac{9}{30} + \frac{162}{270} = \frac{9}{10}$$ Probability of one sale: $$\frac{1}{3}\times \frac{1}{10} + 2\times \frac{2}{3}\times \frac{1}{10}\times \frac{9}{10} = \frac{1}{30} + 2\times \frac{18}{270} = \frac{6}{30} = \frac{1}{5}$$ Probability of two sales: $$\frac{2}{3}\times \frac{1}{10}\times \frac{1}{10} = \frac{2}{300} = \frac{1}{150}$$
03

Calculate the expected value

Now that we have the probabilities for each outcome, we can calculate the expected value of daily sales by multiplying each outcome by its respective probability and summing the results: Expected value = $$0\times \frac{9}{10} + 50{,}000\times \frac{1}{5} + 100{,}000\times \frac{1}{150}$$ Expected value = $$0 + 10{,}000 + \frac{100{,}000}{150}$$ Expected value = $$10{,}000 + \frac{2{,}000}{3} = 10{,}000 + 666.67 \approx 10{,}667$$ Therefore, the expected value of the heavy-equipment salesman's daily sales is approximately $10,667.

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Key Concepts

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

Probability Distribution
A probability distribution is a mathematical function that provides the probabilities of occurrence of different potential outcomes. In this exercise, the probability distribution describes the likelihood of the salesman making different numbers of sales each day.

There are certain key points about probability distributions:
  • They help in understanding how probabilities are spread across different outcomes.
  • Every outcome must have a probability between 0 and 1.
  • The sum of all probabilities for all possible outcomes must equal 1.
The probability distribution here is based on contacts and sales:
  • Contact 1 or 2 customers per day.
  • Make a sale or not with each customer contacted.
For this salesman, the probabilities of contacting customers and making a sale create the framework for calculating the expected sales outcomes.
Sales Outcome
Sales outcomes in this context refer to the possible results of the salesman's efforts in a single day: making no sales, one sale, or two sales. Each outcome is associated with a monetary value, either $0, $50,000, or $100,000.

Calculated probabilities for these outcomes are determined by considering combinations of contacts and sales:
  • No sale: Happens when all contacts result in no sales. This has a high probability because each sale has a small 1/10 chance.
  • One sale: Occur when exactly one contact results in a sale, while others do not. This combines probabilities of one contact being successful and others failing.
  • Two sales: The least likely, as this requires both contacts to be successful, a rarity given the individual low likelihood of a sale.
Understanding sales outcomes helps in predicting performance and planning strategies to improve success rates.
Heavy-Equipment Sales
Heavy-equipment sales are significant because each successful sale nets a large amount of revenue. As shown in this problem, selling a piece of heavy equipment yields $50,000 per unit sold.

This class of sales has unique characteristics:
  • High stakes: Each sale involves large sums of money.
  • Low frequency: Given the high individual cost, the probability of any given contact resulting in a sale is lower.
  • Great impact on earnings: Despite low frequency, successful sales significantly impact daily earnings due to their high value.
Calculating expected value can help sales leaders make informed decisions about where to allocate resources and how to focus sales efforts. For this salesman, determining the expected value of $10,667 provides insight into average daily earnings given typical situation probabilities.

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