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 'or' probability of two non-mutually exclusive events A and B is calculated using the formula: P(A or B) = P(A) + P(B) - P(A and B). For instance, if event A is drawing a red card (P(A) = 0.5) and event B is drawing a queen (P(B) = 0.077), then the probability of drawing either a red card or a queen is 0.5 + 0.077 - P(A and B). Given that P(A and B), the event of drawing a red queen, is 0.0385, the final probability is 0.5385.

Step 1: Understand the Problem

The problem mentions finding the 'or' probabilities for events that are not mutually exclusive. This implies that the events can occur at the same time. The goal is to find the probability that either event A or event B occurs.

Step 2: Identify the Probabilities

Identify the individual probabilities of the two events- event A and event B. For example, if event A is drawing a red card from a deck of 52 cards, its probability, P(A) would be \( \frac {26} {52} = 0.5 \). Similarly, if event B is drawing a queen from the same deck, its probability, P(B) would be \( \frac {4} {52} = 0.077 \).

Step 3: Calculate the Intersection of the Two Events

Calculate the probability that both events occur at the same time, i.e., P(A and B). In our example, this would be the probability that a card drawn is both red and a queen. There are 2 red queens in a deck of 52 cards, so P(A and B) is \( \frac {2} {52} = 0.0385 \).

Step 4: Find the 'or' Probabilities

Use the formula for 'or' probabilities of non-mutually exclusive events which accounts for the possibility of the events overlapping: P(A or B) = P(A) + P(B) - P(A and B). Using the probabilities from the example, P(A or B) = 0.5 + 0.077 - 0.0385 = 0.5385.