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.