We are tasked with finding the value of $\mathbb{P}\left[\frac{B}{A \cup B^C}\right]$, given the following probabilities:

• $\mathbb{P}(A^C) = 0.3$
• $\mathbb{P}(B) = 0.4$
• $\mathbb{P}(A \cap B^C) = 0.5$.

Solution:

Let us proceed step by step.

Step 1: Determine $\mathbb{P}(A)$

We know that: $\mathbb{P}(A) + \mathbb{P}(A^C) = 1$ Substituting $\mathbb{P}(A^C) = 0.3$: $\mathbb{P}(A) = 1 - 0.3 = 0.7$

Step 2: Determine $\mathbb{P}(B^C)$

Similarly, we know: $\mathbb{P}(B) + \mathbb{P}(B^C) = 1$ Substituting $\mathbb{P}(B) = 0.4$: $\mathbb{P}(B^C) = 1 - 0.4 = 0.6$

Step 3: Find $\mathbb{P}(A \cup B^C)$

Using the formula for the union of two events: $\mathbb{P}(A \cup B^C) = \mathbb{P}(A) + \mathbb{P}(B^C) - \mathbb{P}(A \cap B^C)$ Substitute the values:

• $\mathbb{P}(A) = 0.7$
• $\mathbb{P}(B^C) = 0.6$
• $\mathbb{P}(A \cap B^C) = 0.5$ $\mathbb{P}(A \cup B^C) = 0.7 + 0.6 - 0.5 = 0.8$

Step 4: Use the conditional probability formula

The conditional probability formula is: $\mathbb{P}\left[\frac{B}{A \cup B^C}\right] = \frac{\mathbb{P}(B \cap (A \cup B^C))}{\mathbb{P}(A \cup B^C)}$

• The numerator $\mathbb{P}(B \cap (A \cup B^C))$ is simply $\mathbb{P}(B)$, because $B$ is a subset of $A \cup B^C$. $\mathbb{P}(B \cap (A \cup B^C)) = \mathbb{P}(B) = 0.4$
• The denominator $\mathbb{P}(A \cup B^C)$ is already calculated as $0.8$.

Thus: $\mathbb{P}\left[\frac{B}{A \cup B^C}\right] = \frac{0.4}{0.8} = 0.5$

Final Answer:

$\boxed{\frac{1}{2} \text{ (Option B)}}$

Would you like a more detailed explanation of any step? Let me know!

Related Questions:

1. What is the formula for conditional probability, and why does it work?
2. How do you calculate $\mathbb{P}(A \cup B)$ when probabilities overlap?
3. Can the value of $\mathbb{P}(B)$ change depending on $\mathbb{P}(B^C)$? If so, how?
4. What does $\mathbb{P}(A^C)$ mean conceptually in probability theory?
5. How do you interpret conditional probabilities in real-world problems?

Tip:

Always verify the complements ($A^C$ or $B^C$) when solving probability problems to ensure no errors in calculations.