Question

If $A$ and $B$ are two events of a random experiment such that $P(\bar{A}) = 0.3$, $P(B) = 0.4$ and $P(A \cap \bar{B}) = 0.5$, then $P(A \cup B) = \_\_\_\_\_\_\_$

• A. 1
• B. 0.9
• C. 0.5
• D. 0.6

Hint:

Using a Venn diagram makes this very intuitive.
The region $A \cup B$ can be written as the disjoint union of $(A \cap \bar{B})$ and $B$.
Therefore: $P(A \cup B) = P(A \cap \bar{B}) + P(B) = 0.5 + 0.4 = 0.9$.
This requires only a single addition step.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given several probabilities associated with two events $A$ and $B$. We need to compute the probability of the union of the two events, $P(A \cup B)$.

Step 2: Key Formula or Approach:
We will use the fundamental laws of probability:
- Complementary rule: $P(A) = 1 - P(\bar{A})$
- Difference event rule: $P(A \cap \bar{B}) = P(A - B) = P(A) - P(A \cap B)$
- Addition rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Step 3: Detailed Explanation:

• First, calculate the probability of event $A$:
\[ P(A) = 1 - P(\bar{A}) = 1 - 0.3 = 0.7 \]
• Next, use the value of $P(A \cap \bar{B}) = 0.5$ to find the intersection probability $P(A \cap B)$:
\[ P(A \cap \bar{B}) = P(A) - P(A \cap B) \] \[ 0.5 = 0.7 - P(A \cap B) \implies P(A \cap B) = 0.7 - 0.5 = 0.2 \]
• Now, use the addition rule to calculate $P(A \cup B)$:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] \[ P(A \cup B) = 0.7 + 0.4 - 0.2 = 0.9 \]

Step 4: Final Answer:
The probability of the union of the two events $P(A \cup B)$ is 0.9.