Question

If $ A $ and $ B $ are two events such that $ P(A) = 0.7 $, $ P(B) = 0.4 $ and $ P\left( A \cap \overline{B} \right) = 0.5 $, where $\overline{B}$ denotes the complement of $ B $, then $ P\left( B | \left( A \cup \overline{B} \right) \right) $ is equal to

• A. \(\frac{1}{2}\)
• B. \(\frac{1}{4}\)
• C. \(\frac{1}{3}\)
• D. \(\frac{1}{6}\)

Hint:

- Remember \( P(A \cap \overline{B}) = P(A) - P(A \cap B) \) - \( A \cup \overline{B} \) can be visualized using Venn diagrams - For conditional probability \( P(X|Y) \), both numerator and denominator must relate to the same probability space - Simplify complex events using probability identities

Answer 1

The Correct Option is B

To determine \( P\left( B | \left( A \cup \overline{B} \right) \right) \), fundamental probability principles will be applied.

1. First, \( P\left( A \cup \overline{B} \right) \) must be calculated. Using the union formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), all terms are expressed concerning \( \overline{B} \).
2. The expression for \( P(A \cup \overline{B}) \) is:
   \(P(A \cup \overline{B}) = P(A) + P(\overline{B}) - P(A \cap \overline{B})\).
3. Given \( P(A \cap \overline{B}) = 0.5 \) and utilizing \( P(\overline{B}) = 1 - P(B) \), we find:
   \(P(\overline{B}) = 1 - 0.4 = 0.6\).
4. The calculation for \( P(A \cup \overline{B}) \) proceeds as:
   \(P(A \cup \overline{B}) = 0.7 + 0.6 - 0.5 = 0.8\).
5. The conditional probability \( P(B | A \cup \overline{B}) \) is found using the formula:
   \(P(B | A \cup \overline{B}) = \frac{P(B \cap (A \cup \overline{B}))}{P(A \cup \overline{B})}\).
6. Set operations reveal that \( B \cap (A \cup \overline{B}) = (B \cap A) \cup (B \cap \overline{B}) = B \cap A \) since \( B \cap \overline{B} \) is the empty set.
   Consequently, \(P(B \cap (A \cup \overline{B})) = P(B \cap A)\).
7. \( P(B \cap A) \) is calculated as follows:
   \(P(B \cap A) = P(A) - P(A \cap \overline{B}) = 0.7 - 0.5 = 0.2\).
8. Substituting the determined values into the conditional probability formula yields:
   \(P(B | A \cup \overline{B}) = \frac{0.2}{0.8} = \frac{1}{4}\).

Thus, the probability \( P(B | A \cup \overline{B}) \) equals \(\frac{1}{4}\), aligning with the correct answer.