Question

You are given $P(A \cup B) = 0.6$ and $P(A \cup \overline{B}) = 0.8$ then $P(A)$ is

• A. 0.6
• B. 0.2
• C. 0.75
• D. 0.4

Hint:

A useful set identity to remember for these problems is $(A \cup B) \cap (A \cup \overline{B}) = A$. However, this applies to the sets themselves. For probabilities, remember that $P(A \cup B) + P(A \cup \overline{B}) = P(A \cup B \cup \overline{B}) + P((A \cup B) \cap (A \cup \overline{B})) = P(S) + P(A) = 1 + P(A)$.
Using this: $0.6 + 0.8 = 1 + P(A) \implies 1.4 = 1 + P(A) \implies P(A) = 0.4$. This is the fastest way!

Answer 1

The Correct Option is D

We need to find the probability of event A using information about its union with event B and its complement.

Step 1: \color{redWrite the Addition Theorem for both Given Unions
1) $P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.6$.
2) $P(A \cup \overline{B}) = P(A) + P(\overline{B}) - P(A \cap \overline{B}) = 0.8$.

Step 2: \color{redUse the Relationship for Event B
Recall that $P(\overline{B}) = 1 - P(B)$.
Also, recall that $P(A \cap \overline{B}) = P(A) - P(A \cap B)$.

Step 3: \color{redRewrite the Second Equation
Substitute the values from Step 2 into the second union equation:
$0.8 = P(A) + (1 - P(B)) - [P(A) - P(A \cap B)]$.
$0.8 = P(A) + 1 - P(B) - P(A) + P(A \cap B)$.
$0.8 = 1 - P(B) + P(A \cap B)$.
Rearranging this gives: $P(B) - P(A \cap B) = 1 - 0.8 = 0.2$.

Step 4: \color{redFinal Substitution to find P(A)
Now, take the first union equation from
Step 1:
$P(A) + [P(B) - P(A \cap B)] = 0.6$.
Substitute the value $(P(B) - P(A \cap B)) = 0.2$ found in
Step 3:
$P(A) + 0.2 = 0.6$.
$P(A) = 0.6 - 0.2 = 0.4$.
The probability $P(A)$ is 0.4, which matches Option (4).