Problem 61 - College Algebra

Question

Explain how to find or probabilities with events that are not mutually exclusive. Give an example.

Step-by-Step Solution

Answer

The probability of either of two events happening, where the events are not mutually exclusive (meaning they can both occur), is computed using the formula \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). For an example where out of 30 students, 15 love football, 10 love music, and 5 love both, the probability that a randomly chosen student loves either football or music (or both) is \(\frac{2}{3}\).

Step 1: Understanding the notion of 'or probabilities' for events that are not mutually exclusive

In probability theory, 'or' typically means that either one event, the other, or both can occur. For events that are not mutually exclusive, meaning they can occur at the same time, the probability of either of them happening is calculated with the formula: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). This formula accounts for the overlap between A and B, the area where both events can happen.

Step 2: Applying the understanding to an example

Suppose there are 30 students in a class: 15 love football (Event A), 10 love music (Event B) and 5 love both (overlap of events A and B). The probability that a randomly chosen student loves either football or music (or both), using the formula above, is calculated as: \(P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{15}{30} + \frac{10}{30} - \frac{5}{30} = \frac{20}{30} = \frac{2}{3}\). So, the probability that a student loves football or music (or both) is \(\frac{2}{3}\).