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This image presents a problem in probability, specifically using Bayes' theorem. Here's a breakdown of the solution:

### Problem Overview
We are given:
- \( P(R) = 0.1 \): The probability of event \( R \).
- \( P(P | R) = 0.2 \): The probability of event \( P \) given that \( R \) has occurred.
- \( P(P | \neg R) = 0.05 \): The probability of event \( P \) given that \( R \) has not occurred.
- \( P(\neg R) = 0.9 \): The probability of \( R \) not occurring.

The goal is to find \( P(R | P) \), the probability of \( R \) given \( P \) has occurred.

### Solution Steps
1. **Calculate \( P(P) \) Using the Law of Total Probability:**

   The law of total probability states:
   \[
   P(P) = P(P | R) \cdot P(R) + P(P | \neg R) \cdot P(\neg R)
   \]
   Substituting the given values:
   \[
   P(P) = (0.2 \cdot 0.1) + (0.05 \cdot 0.9) = 0.02 + 0.045 = 0.065
   \]

2. **Apply Bayes' Theorem to Find \( P(R | P) \):**

   Bayes' theorem is given by:
   \[
   P(R | P) = \frac{P(P | R) \cdot P(R)}{P(P)}
   \]
   Plugging in the values:
   \[
   P(R | P) = \frac{0.2 \cdot 0.1}{0.065} = \frac{0.02}{0.065} \approx 0.308
   \]
   Thus, \( P(R | P) \approx 0.308 \) or 30.8%.

### Summary
The solution uses Bayes' theorem to find the posterior probability \( P(R | P) \), which is approximately 30.8%.

Would you like further clarification on any step or additional details?

Here are some related questions for practice:
1. How would you calculate \( P(\neg R | P) \) using similar principles?
2. What would happen if \( P(P | R) \) or \( P(P | \neg R) \) were different?
3. How does changing \( P(R) \) affect \( P(R | P) \)?
4. Can Bayes' theorem be used if events are independent?
5. What is the importance of the law of total probability in this calculation?

**Tip:** Bayes' theorem is a powerful tool for updating probabilities based on new information.

Given probability values P(R) = 0.1, P(P | R) = 0.2, P(P | ¬R) = 0.05, and P(¬R) = 0.9, calculate P(R | P) using Bayes' theorem.

Learn how to use Bayes' theorem to calculate the probability P(R | P) with given probabilities P(R), P(P | R), P(P | ¬R), and P(¬R). This solution demonstrates key probability concepts, including conditional probability and the law of total probability. Suitable for students in Grades 11-12.

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