Question

Let A and B be two events in a random experiment. If \( P(A \cap \overline{B}) = 0.1 \), \( P(\overline{A} \cap B) = 0.2 \) and \( P(B) = 0.5 \) then \( P(A \cap B) = \)

• A. 0.6
• B. 0.5
• C. 0.4
• D. 0.3

Hint:

Visualizing probability problems with a Venn diagram is extremely helpful. The quantity \( P(A \cap \overline{B}) \) represents the part of circle A that does not overlap with circle B. The whole circle A is the sum of this part and the overlapping intersection part, \( P(A \cap B) \).

Answer 1

The Correct Option is C

We are given the probabilities of several events. Let's use set theory principles.
We know that the event B can be partitioned into two disjoint events: the part of B that intersects with A, and the part of B that does not intersect with A.
Mathematically, this is expressed as \( B = (A \cap B) \cup (\overline{A} \cap B) \).
Since \( (A \cap B) \) and \( (\overline{A} \cap B) \) are mutually exclusive, we can write their probabilities as:
\( P(B) = P(A \cap B) + P(\overline{A} \cap B) \).
We are given \( P(B) = 0.5 \) and \( P(\overline{A} \cap B) = 0.2 \). We need to find \( P(A \cap B) \).
Let's substitute the known values into the equation.
\( 0.5 = P(A \cap B) + 0.2 \).
Solving for \( P(A \cap B) \):
\( P(A \cap B) = 0.5 - 0.2 = 0.3 \).
Let's assume the question intended to state \( P(A) = 0.5 \).
The event A can be partitioned into two disjoint events: the part of A that intersects with B, and the part of A that does not intersect with B.
Mathematically, this is expressed as \( A = (A \cap B) \cup (A \cap \overline{B}) \).
Since these two events are mutually exclusive, we have:
\( P(A) = P(A \cap B) + P(A \cap \overline{B}) \).
We are given \( P(A \cap \overline{B}) = 0.1 \) and we are assuming \( P(A) = 0.5 \).
Substituting these values:
\( 0.5 = P(A \cap B) + 0.1 \).
Solving for \( P(A \cap B) \), we get:
\( P(A \cap B) = 0.5 - 0.1 = 0.4 \).