To solve this problem, let's break it down step by step:

Given:

• $P(A) = 0.35$, $P(B) = 0.45$, $P(C) = 0.2$
• $P(X|A) = 0.8$, $P(X|B) = 0.65$, $P(X|C) = 0.3$

We are tasked with finding:

1. $P(X)$ (the total probability of $X$).
2. $P(B|X)$ (the conditional probability of $B$ given $X$).

Step 1: Using the Total Probability Theorem to Find $P(X)$

The total probability theorem states: $P(X) = P(A)P(X|A) + P(B)P(X|B) + P(C)P(X|C)$

Substitute the given values: $P(X) = (0.35 \times 0.8) + (0.45 \times 0.65) + (0.2 \times 0.3)$

Calculate: $P(X) = 0.28 + 0.2925 + 0.06$

$P(X) = 0.6325$

Step 2: Finding $P(B|X)$ Using Bayes' Theorem

Bayes' theorem states: $P(B|X) = \frac{P(B)P(X|B)}{P(X)}$

Substitute the known values: $P(B|X) = \frac{0.45 \times 0.65}{0.6325}$

Calculate: $P(B|X) = \frac{0.2925}{0.6325}$

$P(B|X) \approx 0.4624$

Final Answers:

1. $P(X) = 0.6325$
2. $P(B|X) \approx 0.4624$

Would you like a more detailed explanation of any step or related questions? Here are five related questions for further exploration:

1. How would $P(X)$ change if $P(C)$ were increased?
2. Can we find $P(A|X)$ and $P(C|X)$ using similar calculations?
3. What is the significance of $P(X|A)$, $P(X|B)$, and $P(X|C)$ in real-world applications?
4. How does the total probability theorem apply in scenarios with more than three events?
5. Can you determine the likelihood of $X$ if only $A$ or $C$ occurs?

Tip: Always double-check probabilities to ensure they add up correctly when using the total probability theorem