Problem 13 - A First Course in Probability

Question

A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability \(.3,\) and his second will lead independently to a sale with probability .6. Any sale made is equally likely to be either for the deluxe model, which costs \(\$ 1000\), or the standard model, which costs \(\$ 500 .\) Determine the probability mass function of \(X,\) the total dollar value of all sales.

Step-by-Step Solution

Answer

The probability mass function of the total dollar value of all sales (\(X\)) can be represented as: \(P(X=k)= \begin{cases} 0.28 & \text{if } k=0, \\ 0.27 & \text{if } k=500, \\ 0.27 & \text{if } k=1000, \\ 0.09 & \text{if } k=1500, \\ 0.045 & \text{if } k=2000. \end{cases}\)

Step 1: Identify the variables and their possible values

Let's denote the two appointments by \(A\) and \(B\). Appointment \(A\) has a probability of sale of \(0.3\) and appointment \(B\) has a probability of sale of \(0.6\). The possible sales values are \(\$1000\) for the deluxe model and \(\$500\) for the standard model. The total sales value, \(X\), can take possible values of \(\$0\), \(\$500\), \(\$1000\), \(\$1500\), and \(\$2000\).

Step 2: Calculate probabilities for different scenarios

We will now consider the possible scenarios of sales and calculate the probabilities for each scenario: 1. No Sales: Probability of no sale at both appointment \(A\) and \(B\) is \((1-0.3)(1-0.6)=0.7 \times 0.4=0.28\) 2. One Sale of standard model or deluxe model - At appointment \(A\) (\(0.3\) probability of sale) with a \(0.5\) probability of standard model and a 0.5 probability of deluxe model. - At appointment \(B\) (\(0.6\) probability of sale) with a \(0.5\) probability of standard model and a 0.5 probability of deluxe model. 3. Two Sales - Both the standard model: Probability is \(0.3 \times 0.5 \times 0.6 \times 0.5 = 0.045\) - Both the deluxe model: Probability is \(0.3 \times 0.5 \times 0.6 \times 0.5 = 0.045\) - One standard and one deluxe model: Probability is \(0.3 \times 0.5 \times 0.6 \times 0.5 + 0.3 \times 0.5 \times 0.6 \times 0.5 = 0.09\)

Step 3: Calculate the probability of each sales value

Using the probabilities calculated for different scenarios, we can now calculate the probabilities for each possible sales value: 1. \(X=0\) (No Sales): Probability is \(0.28\) 2. \(X=500\) (One Sale of standard model): Probability is \(0.3 \times 0.5 \times 0.4 + 0.6 \times 0.5 \times 0.7 = 0.27\) 3. \(X=1000\) (One Sale of deluxe model): Probability is \(0.3 \times 0.5 \times 0.4 + 0.6 \times 0.5 \times 0.7 = 0.27\) 4. \(X=1500\) (One standard and one deluxe model): Probability is \(0.09\) 5. \(X=2000\) (Two Sales of deluxe model): Probability is \(0.045\)

Step 4: Write the probability mass function

Now, we write the probability mass function of \(X\), denoted by \(P(X=k)\), for each sales value: 1. \(P(X=0)=0.28\) 2. \(P(X=500)=0.27\) 3. \(P(X=1000)=0.27\) 4. \(P(X=1500)=0.09\) 5. \(P(X=2000)=0.045\) So, the probability mass function of the total dollar value of all sales is as follows: $P(X=k)= \begin{cases} 0.28 & \text{if } k=0, \\ 0.27 & \text{if } k=500, \\ 0.27 & \text{if } k=1000, \\ 0.09 & \text{if } k=1500, \\ 0.045 & \text{if } k=2000. \end{cases}$

Key Concepts

Sales Probability

When we discuss sales probability, we're referring to the likelihood that an event, such as making a sale, will occur. In the context of our problem, a salesman has two separate opportunities to sell encyclopedias, with different probabilities associated with each appointment. Understanding sales probability involves not only the likelihood of a sale occurring but also comprehending the specific outcomes associated with each sale.

For instance, the first appointment has a 30% chance of leading to a sale, while the second has a higher probability of 60%. It's crucial to note that these events are independent, meaning the outcome of one does not affect the outcome of the other. Furthermore, each sale could result in either a standard or deluxe model purchase, introducing another layer of probability to consider. By calculating these probabilities, a business can forecast sales and make informed decisions.

Probability Calculations

Calculating probabilities involves determining the likelihood of various possible outcomes. In our scenario, to find the probability mass function of the total sales value, we need to consider all potential scenarios, from making no sales at all to selling two deluxe models.

The process begins with identifying variables, their potential outcomes, and the associated probabilities. Then, we calculate the probability of each outcome. Some scenarios, especially those involving multiple events, might require multiplication of individual probabilities, adhering to the rules of probability for independent events.

For instance, to compute the probability of selling one standard model from each appointment, we'd multiply the probability of selling a standard model at the first appointment by the probability of selling a standard model at the second appointment. Ensuring accuracy in these calculations is paramount because slight errors can significantly distort the probability distribution and subsequent decisions based on these calculations.

Discrete Random Variables

Discrete random variables, like the total dollar value of sales in our problem, can take on only a specific set of values, often integers or counts. Every discrete random variable has a probability mass function, which gives the probability that the random variable equals each of its possible values.

In simpler terms, it tells us how likely it is that the salesperson will make a certain amount of money from selling encyclopedias. We represent it by mapping each possible value to its probability. For example, the probability mass function tells us that the probability of making no sales (\(X=0\)) is 0.28, while the probability of making exactly \$2000 from sales (\(X=2000\)) is 0.045. The sum of all probabilities in a probability mass function should be 1 since one of the possible outcomes must occur.

The discrete nature of this variable is crucial to our understanding and calculation methods since the continuous probability distributions operate on a different set of rules and calculations.