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 \(X\), the total dollar value of all sales, is: $P(X) = \begin{cases} 0.28 & \text{ if } X = 0 \\ 0.27 & \text{ if } X = 500 \\ 0.315 & \text{ if } X = 1000 \\ 0.09 & \text{ if } X = 1500 \\ 0.045 & \text{ if } X = 2000 \\ 0 & \text{ otherwise} \end{cases}$

Step 1: List possible sales outcomes

To begin, let's list the possible outcomes for the two appointments: 1. No sales in both appointments 2. A sale in the first appointment, but no sale in the second appointment 3. No sale in the first appointment, but a sale in the second appointment 4. Sales in both appointments

Step 2: Determine probability and dollar value for each outcome

Next, we will determine the probability and total dollar value for each of these outcomes: 1. No sales in both appointments: Probability is \((1 - 0.3)(1 - 0.6) = 0.7 \times 0.4 = 0.28\). In this case, the total dollar value is \(0\). 2. A sale in the first appointment, but no sale in the second appointment: Probability is \(0.3 \times 0.4 = 0.12\). In this case, the total dollar value can be either \(500\) or \(1000\). Both of these outcomes are equally likely. So, the probability for the total dollar value of \(500\) is \(0.12 \times 0.5 = 0.06\), and the probability for the total dollar value of \(1000\) is also \(0.12 \times 0.5 = 0.06\). 3. No sale in the first appointment, but a sale in the second appointment: Probability is \(0.7 \times 0.6 = 0.42\). In this case, the total dollar value can be either \(500\) or \(1000\). Both of these outcomes are equally likely. So, the probability for the total dollar value of \(500\) is \(0.42 \times 0.5 = 0.21\), and the probability for the total dollar value of \(1000\) is also \(0.42 \times 0.5 = 0.21\). 4. Sales in both appointments: Probability is \(0.3 \times 0.6 = 0.18\). In this case, there are two possible outcomes for both sales: either both deluxe (total dollar value of \(2000\)), both standard (total dollar value of \(1000\)), or one of each (total dollar value of \(1500\)). To simplify further, let's calculate the probabilities of each of these possibilities. a) Both deluxe: Probability is \(0.18 \times 0.5 \times 0.5 = 0.045\) b) Both standard: Probability is \(0.18 \times 0.5 \times 0.5 = 0.045\) c) One of each deluxe and standard (order doesn't matter): Probability is \(0.18 \times 0.5 \times 0.5 \times 2 = 0.09\)

Step 3: Determine the probability mass function of X

Now, we will determine the probability mass function of \(X\) using the probabilities and total dollar values calculated in step 2. The probability mass function is as follows: - \(P(X = 0) = 0.28\) - \(P(X = 500) = 0.06 + 0.21 = 0.27\) - \(P(X = 1000) = 0.06 + 0.21 + 0.045 = 0.315\) - \(P(X = 1500) = 0.09\) - \(P(X = 2000) = 0.045\) So, the probability mass function of \(X\) is: $P(X) = \begin{cases} 0.28 & \text{ if } X = 0 \\ 0.27 & \text{ if } X = 500 \\ 0.315 & \text{ if } X = 1000 \\ 0.09 & \text{ if } X = 1500 \\ 0.045 & \text{ if } X = 2000 \\ 0 & \text{ otherwise} \end{cases}$

Key Concepts

Probability Theory

Probability theory is a fascinating and essential field of mathematics
that allows us to quantify and analyze uncertainty. It helps answer questions about how likely an event is to occur. In essence, probability measures the likelihood of an event happening out of the total number of possibilities.

Some key aspects of probability theory include:

• Experiments and Sample Spaces: Probability considers the set of all possible outcomes, called the sample space.
• Events: Particular outcomes or sets of outcomes of an experiment are called events.
• Probability of Events: Each event is assigned a probability, a value between 0 (impossible) and 1 (certain).

Using simple rules, like multiplication for independent events, probability theory can accurately predict the likelihood of different scenarios, as you saw in the original exercise with the sales outcomes.

This theory underpins much of modern statistics and is used in fields ranging from science to finance.

Random Variables

In probability theory, random variables are fundamental concepts used to represent numerical outcomes associated with random phenomena.
They serve as a bridge between the abstract world of probabilities and the concrete realm of numbers.

Here's what you should know about random variables:

• Definition: A random variable assigns a numerical value to each outcome in a sample space. They can be discrete or continuous.
• Discrete Random Variables: These are random variables that take specific, countable values. In the example of the salesman, the random variable represents the total sales amount.
• Probability Distribution: For discrete random variables, the probability distribution is expressed by a probability mass function (PMF).
• PMF: A PMF assigns probabilities to each possible value that the random variable can take, as was done in calculating the probabilities of different sales amounts.

This concept is crucial in helping to model real-world scenarios where outcomes are not deterministically predictable.

Discrete Distributions

Among the many types of probability distributions, discrete distributions are vital in handling situations where outcomes take distinct separate values.
They are especially useful for modeling scenarios involving counts or individually distinct categories.

Key aspects of discrete distributions include:

• Countable Outcomes: Unlike continuous distributions, discrete distributions deal with countable outcomes, such as the number of sales.
• Examples: Some common discrete distributions include the binomial distribution, Poisson distribution, and the geometric distribution.
• Probability Mass Function (PMF): The PMF is a defining function of a discrete distribution, indicating each possible outcome's probability, as seen in the calculation of total dollar sales in appointments.

By understanding discrete distributions, we can effectively predict and analyze data in fields like business, which involves sales figures, stock counts, or customer visits. They are essential for decision-making processes based on quantitative data.