39. You are given that P(A) = 0.15 and P(B) = 0.40. Do you have enough information to find P(A or B)? Explain.

Verified step by step guidance

1. Step 1: Recall the formula for the probability of the union of two events, P(A or B), which is given by P(A ∪ B) = P(A) + P(B) - P(A ∩ B). This formula accounts for the overlap between events A and B to avoid double-counting.
2. Step 2: Substitute the given values into the formula. You are provided P(A) = 0.15 and P(B) = 0.40. However, the formula also requires the value of P(A ∩ B), the probability that both events A and B occur simultaneously.
3. Step 3: Assess whether P(A ∩ B) is provided or can be inferred. In this problem, P(A ∩ B) is not given, and there is no information about whether events A and B are independent or dependent. Without P(A ∩ B), you cannot calculate P(A or B) precisely.
4. Step 4: If events A and B were independent, you could calculate P(A ∩ B) using the formula P(A ∩ B) = P(A) × P(B). However, independence is not stated in the problem, so you cannot assume it.
5. Step 5: Conclude that you do not have enough information to find P(A or B) because the value of P(A ∩ B) is missing, and the relationship between events A and B (independent or dependent) is not specified.

Key Concepts

Probability of Events

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In this context, P(A) and P(B) represent the probabilities of events A and B occurring, respectively. Understanding how to interpret these probabilities is essential for analyzing the relationship between the events.

Union of Events

The union of two events, denoted as P(A or B), represents the probability that at least one of the events occurs. To calculate this, we typically use the formula P(A or B) = P(A) + P(B) - P(A and B). However, without knowing P(A and B), we cannot determine P(A or B) accurately.

Independence of Events

Events A and B are independent if the occurrence of one does not affect the probability of the other. If A and B are independent, we can calculate P(A and B) as P(A) * P(B). However, the question does not provide information about their independence, which is crucial for determining P(A or B) without additional data.