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.