Problem 21 - A First Course in Probability

Question

A total of 500 married working couples were polled about their annual salaries, with the following information resulting: $$\begin{array}{lcc} \hline & \multicolumn{2}{c} {\text { Husband }} \\ \\)\cline { 2 - 3 }\\( \text { Wife } & \begin{array}{c} \text { Less than } \\ \$ 25,000 \end{array} & \begin{array}{c} \text { More than } \\ \$ 25,000 \end{array} \\ \hline \text { Less than \$25,000 } & 212 & 198 \\ \text { More than \$25,000 } & 36 & 54 \\ \hline \end{array}$$ For instance, in 36 of the couples, the wife earned more and the husband earned less than \(\$ 25,000 .\) If one of the couples is randomly chosen, what is (a) the probability that the husband earns less than \(\$ 25,000 ?\) (b) the conditional probability that the wife earns more than \(\$ 25,000\) given that the husband earns more than this amount? (c) the conditional probability that the wife earns more than \(\$ 25,000\) given that the husband earns less than this amount?

Step-by-Step Solution

Answer

(a) The probability that the husband earns less than $25,000 is \(0.496\). (b) The conditional probability that the wife earns more than $25,000 given that the husband earns more than this amount is \(0.2143\). (c) The conditional probability that the wife earns more than $25,000 given that the husband earns less than this amount is \(0.1452\).

Step 1: Find the Total Number of Couples

We can find the total number of couples by adding the values in the table. Notice that the problem statement mentions a total of 500 married working couples, so we don't need to perform computations ourselves. Total couples: 500

Step 2: Calculate the Probability That a Husband Earns Less Than \(25,000

To find this probability, we need to determine how many husbands earn less than \)25,000 in the table and divide by the total number of couples. In the table, we can see that 212 + 36 husbands earn less than $25,000. Number of husbands earning less than $25,000: 212 + 36 = 248 Probability = \(\frac{\text{Number of husbands earning less than $25,000}}{\text{Total couples}} = \frac{248}{500} = 0.496\)

Step 3: Calculate the Conditional Probability That a Wife Earns More Than \(25,000 Given the Husband Earns More Than This Amount

Given that the husband earns more than \)25,000, we can rule out the first column of the table. We now want to find the probability that the wife earns more than $25,000 within that group. First, we need to find the number of couples where the husband earns more than $25,000. Number of husbands earning more than $25,000: 198 + 54 = 252 Next, find the number of couples where the wife earns more than $25,000 given that the husband earns more than this amount (54 couples according to the table). Conditional Probability = \(\frac{\text{Couples where both earn more than \(25,000}}{\text{Couples where husband earns more than \)25,000}} = \frac{54}{252} = 0.2143\)

Step 4: Calculate the Conditional Probability That a Wife Earns More Than \(25,000 Given the Husband Earns Less Than This Amount

Given that the husband earns less than \)25,000, we can rule out the second column of the table. We now want to find the probability that the wife earns more than $25,000 within that group. First, we need to find the number of couples where the husband earns less than $25,000, which we did in Step 2 (248 couples). Next, find the number of couples where the wife earns more than $25,000 given that the husband earns less than this amount (36 couples according to the table). Conditional Probability = \(\frac{\text{Couples where wife earns more than \(25,000 and husband earns less}}{\text{Couples where husband earns less than \)25,000}} = \frac{36}{248} = 0.1452\) In conclusion, we've found the probabilities asked for in the exercise: (a) The probability that the husband earns less than $25,000 is 0.496. (b) The conditional probability that the wife earns more than $25,000 given that the husband earns more than this amount is 0.2143. (c) The conditional probability that the wife earns more than $25,000 given that the husband earns less than this amount is 0.1452.

Key Concepts

Probability Theory

Probability Theory is the fundamental framework used to evaluate the likelihood of various outcomes. It serves as a guiding principle in many fields, like statistics, finance, and science, helping us understand chance and uncertainty. Here’s a straightforward way to understand the core of probability theory:

• Probability is a number between 0 and 1. A probability of 0 means an event will not occur, while a probability of 1 means it is certain to happen.
• The probability of an event is determined by the formula: \[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
• When you add up the probabilities of all possible events in a sample space, it will sum up to 1.

In the exercise, probability is used to predict how likely it is for a husband or a wife to earn less or more than $25,000. Each outcome we examine stems from calculating the proportion of couples fitting the criteria against the total number of couples.

Joint Probability

Joint Probability measures the likelihood of two events happening at the same time. It's a fundamental concept when we're interested in understanding the relationships between events:

• The joint probability of events A and B is written as \( P(A \cap B) \), meaning the chance of both A and B occurring together.
• It requires a consideration of how two events are related in a grid-like space, where rows and columns represent different outcomes.
• Joint probability is calculated using: \[ P(A \cap B) = \frac{\text{Number of favorable outcomes for both A and B}}{\text{Total number of possible outcomes}} \]

In this exercise, joint probability comes into play when considering both the husband and wife’s salaries together, illustrating how two different events (in this case, salary levels) can intersect in a data table to form a new probability scenario.

Independent Events

Independent events are those where the occurrence of one event does not affect the occurrence of the other. This concept is critical in probability, as it simplifies the way we calculate joint events:

• Two events, A and B, are independent if and only if: \[ P(A \cap B) = P(A) \times P(B) \]
• This formula indicates that the chance of both events occurring is simply the product of their individual probabilities.
• Knowledge of one event occurring tells us nothing about the other event.

In the given exercise, husbands' and wives' earnings potential isn’t presented as independent since the calculation involves conditional probability instead of multiplying standalone probabilities. Events are dependent when the occurrence or monthly income of one individual might impact the other, dictated by their joint lifestyle choices or careers.