Q.You are given that $A$ and $B$ are two events such that $P(B) = \dfrac{3}{5}$, $P(A \mid B) = \dfrac{1}{2}$ and $P(A \cup B) = \dfrac{4}{5}$, then $P(A)$ equals
(A) $\dfrac{3}{10}$
(B) $\dfrac{1}{5}$
(C) $\dfrac{1}{2}$
(D) $\dfrac{3}{5}$

Concept: Conditional Probability — $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$.

Step 1: Use the conditional probability formula to find $P(A \cap B)$.

So $P(A \cap B) = \frac{1}{2} \cdot \frac{3}{5} = \frac{3}{10}$.

Step 2: Apply the addition rule:

Substitute known values:

Step 3: Solve for $P(A)$: …

We start with what we know: $P(B) = \frac{3}{5}$, $P(A \mid B) = \frac{1}{2}$, and $P(A \cup B) = \frac{4}{5}$. The goal is to find $P(A)$.

The key idea is that conditional probability gives us a direct link between $P(A \mid B)$ and the intersection $P(A \cap B)$. Once we have the intersection, the union formula lets us solve for $P(A)$.

1. Find $P(A \cap B)$ using conditional probability. By definition, $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$. Substitute the given values:

Multiply both sides by $\frac{3}{5}$:

2. Use the inclusion–exclusion principle for $P(A \cup B)$. The formula is:

Plug in what we have:

3. Solve for $P(A)$. First, combine the constants on the right:

So the equation becomes:

$\frac{4}{5} = P(A) + \frac{3}{10}$ …

Method: Solving for an unknown marginal probability

Use this when one marginal (say $P(A)$) is the unknown, and you are given a conditional probability plus a union.

Steps

Step 1: Turn the conditional into the intersection.

Step 2: Write the addition rule as an equation in the unknown.

Every quantity except $P(A)$ is now known.

Step 3: Rearrange and solve.

$P(A)=P(A\cup B)-P(B)+P(A\cap B).$ …

Common Mistakes

Mistake 1: Setting $P(A\cap B)$ equal to $P(A\mid B)$.

Why it's wrong: the conditional must be multiplied by $P(B)$ to become a joint probability. Correct approach: $P(A\cap B)=P(A\mid B)\,P(B)$.

Mistake 2: Sign errors when isolating $P(A)$. …