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 solve the problem of finding \( P\left( B | \left( A \cup \overline{B} \right) \right) \), we will use some basic probability principles and formulas.

1. Initially, we need to determine \( P\left( A \cup \overline{B} \right) \). Using the formula for the probability of the union of events \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), we first need to express all terms in terms of \( \overline{B} \).
2. We can express \( P(A \cup \overline{B}) \) as:
   \(P(A \cup \overline{B}) = P(A) + P(\overline{B}) - P(A \cap \overline{B})\).
3. Given \( P(A \cap \overline{B}) = 0.5 \) and using the fact that \( P(\overline{B}) = 1 - P(B) \), we have:
   \(P(\overline{B}) = 1 - 0.4 = 0.6\).
4. So, calculate \( P(A \cup \overline{B}) \):
   \(P(A \cup \overline{B}) = 0.7 + 0.6 - 0.5 = 0.8\).
5. Now, to find the conditional probability \( P(B | A \cup \overline{B}) \), use the conditional probability formula:
   \(P(B | A \cup \overline{B}) = \frac{P(B \cap (A \cup \overline{B}))}{P(A \cup \overline{B})}\).
6. We know from set operations that \( B \cap (A \cup \overline{B}) = (B \cap A) \cup (B \cap \overline{B}) = B \cap A \) because \( B \cap \overline{B} \) is an empty set.
   Therefore, \(P(B \cap (A \cup \overline{B})) = P(B \cap A)\).
7. To find \( P(B \cap A) \):
   \(P(B \cap A) = P(A) - P(A \cap \overline{B}) = 0.7 - 0.5 = 0.2\).
8. Now substitute the known values into the conditional probability formula:
   \(P(B | A \cup \overline{B}) = \frac{0.2}{0.8} = \frac{1}{4}\).

Therefore, the probability \( P(B | A \cup \overline{B}) \) is \(\frac{1}{4}\), which matches the correct answer option.

Answer 2

Step 1: Find \( P(A \cap B) \)
\[ P(A \cap \overline{B}) = P(A) - P(A \cap B) \] \[ 0.5 = 0.7 - P(A \cap B) \] \[ P(A \cap B) = 0.2 \]

Step 2: Calculate \( P(A \cup \overline{B}) \)
Using probability rules: \[ P(A \cup \overline{B}) = P(A) + P(\overline{B}) - P(A \cap \overline{B}) \] \[ = 0.7 + (1 - 0.4) - 0.5 \] \[ = 0.7 + 0.6 - 0.5 = 0.8 \]

Step 3: Find \( P(B \cap (A \cup \overline{B})) \)
\[ P(B \cap (A \cup \overline{B})) = P((B \cap A) \cup (B \cap \overline{B})) \] \[ = P(A \cap B) + P(\emptyset) = 0.2 + 0 = 0.2 \]

Step 4: Compute conditional probability
\[ P(B | A \cup \overline{B}) = \frac{P(B \cap (A \cup \overline{B}))}{P(A \cup \overline{B})} = \frac{0.2}{0.8} = \frac{1}{4} \]