Реши

To solve this problem, we need to use the concept of probability trees and the law of total probability. Here’s a step-by-step explanation: ### Problem Summary We have an experiment with the following events: - Event A occurs with a probability of 0.3. - If event A occurs, event C occurs with a probability of 0.2. - If event A does not occur (event \( \overline{A} \)), event C occurs with a probability of 0.4. We need to find the total probability of event C occurring. ### Steps to Solve 1. **Identify the Given Probabilities** - \( P(A) = 0.3 \) - \( P(C | A) = 0.2 \) (Probability of C given A occurs) - \( P(\overline{A}) = 1 - P(A) = 0.7 \) (Probability of A not occurring) - \( P(C | \overline{A}) = 0.4 \) (Probability of C given A does not occur) 2. **Use the Law of Total Probability** The total probability of C, \( P(C) \), can be found using: \[ P(C) = P(C \cap A) + P(C \cap \overline{A}) \] where: \[ P(C \cap A) = P(C | A) \times P(A) \] \[ P(C \cap \overline{A}) = P(C | \overline{A}) \times P(\overline{A}) \] 3. **Calculate Each Probability Term** - Calculate \( P(C \cap A) \): \[ P(C \cap A) = 0.2 \times 0.3 = 0.06 \] - Calculate \( P(C \cap \overline{A}) \): \[ P(C \cap \overline{A}) = 0.4 \times 0.7 = 0.28 \] 4. **Calculate Total Probability of C** \[ P(C) = 0.06 + 0.28 = 0.34 \] Therefore, the total probability of event C occurring is **0.34**.