Question

Employment data at a large company reveal that \(72 \%\) of the workers are married, that \(44 \%\) are college graduates, and that half of the college grads are married. What's the probability that a randomly chosen worker a) is neither married nor a college graduate? b) is married but not a college graduate? c) is married or a college graduate?

Short answer

a) 0.06, b) 0.50, c) 0.94

Step-by-step solution

Define Events

Let \( M \) represent the event that a worker is married, and \( G \) represent the event that a worker is a college graduate. From the problem statement, we know that \( P(M) = 0.72 \), \( P(G) = 0.44 \), and \( P(M \cap G) = 0.22 \) since half of the college grads are married.

Calculate Probability of Neither Married Nor College Graduate

To find the probability that a worker is neither married nor a college graduate, use the complement rule: \( P(\text{neither } M \text{ nor } G) = 1 - P(M \cup G) \). First, calculate \( P(M \cup G) \) using the formula \( P(M \cup G) = P(M) + P(G) - P(M \cap G) = 0.72 + 0.44 - 0.22 = 0.94 \). Thus, \( P(\text{neither } M \text{ nor } G) = 1 - 0.94 = 0.06 \).

Calculate Probability of Married but Not a College Graduate

The probability that a worker is married but not a college graduate can be found using \( P(M \cap \overline{G}) = P(M) - P(M \cap G) \). Therefore, \( P(M \cap \overline{G}) = 0.72 - 0.22 = 0.50 \).

Calculate Probability of Married or a College Graduate

To find the probability that a worker is married or a college graduate, use the result from Step 2, \( P(M \cup G) = 0.94 \). This was calculated as \( 0.72 + 0.44 - 0.22 \). So, \( P(M \cup G) = 0.94 \).

Key concepts

Complement Rule

Understanding the Complement Rule in probability theory is essential for calculating the likelihood of an event not happening. The complement of an event consists of all possible outcomes that are not part of the original event. If you know the probability of an event, you can easily determine the probability of its complement using the formula:\[ P(A^c) = 1 - P(A) \]where \( A^c \) is the complement of event \( A \), and \( P(A) \) is the probability of event \( A \). In the case of our exercise, to find the probability of a worker being neither married nor a college graduate, we first calculate the probability of a worker being either married or a college graduate, and then subtract this from 1.

Event Probability

The probability of an event is a measure that quantifies the likelihood that the event will occur. This measure ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty. For example, in our problem, the event \( M \) (the worker being married) has a probability of \( P(M) = 0.72 \), meaning that there is a 72% chance of randomly picking a married worker. Similarly, the event \( G \) (the worker being a college graduate) has a probability of \( P(G) = 0.44 \).Calculating event probabilities often involves identifying all possible outcomes and how they pertain to your specific event. Understanding them helps solve more complex probabilities, such as combined events like the intersection or union of multiple events.

Intersection of Events

When two or more events occur simultaneously, they form what is called the "intersection" of events. This is symbolized as \( A \cap B \) and represents the set of outcomes common to both events \( A \) and \( B \). In probability terms, it is described as \( P(A \cap B) \). For our problem, the intersection of being married and being a college graduate is represented as \( P(M \cap G) = 0.22 \), indicating that 22% of the company's workers are both married and college graduates. Knowing the intersection is crucial when using principles like the complement rule, as it helps calculate the unions and assists in solving probability tasks involving multiple events.

Union of Events

Unions in probability involve finding the probability that at least one of multiple events will occur. It's symbolized as \( A \cup B \) and includes all outcomes that are in \( A \) or \( B \), or in both.To calculate the probability of the union of two events, you add their individual probabilities and subtract the probability of their intersection, avoiding double-counting the overlapping outcomes. \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]For instance, for the problem, the probability that a worker is either married or a college graduate (or both) is denoted as \( P(M \cup G) = 0.94 \), calculated by adding \( P(M) = 0.72 \) and \( P(G) = 0.44 \), then subtracting \( P(M \cap G) = 0.22 \). Understanding unions is vital in solving parts of the exercise that involve calculating inclusive probabilities.