If P(A) = 0.4, P(B) = 0.3, P(A∩B) = 0.1, find P(A|B) using Bayes theorem

Question

If $P(A) = 0.4$ , $P(B) = 0.3$ , and $P(A \cap B) = 0.1$ , find $P(A|B)$ .

(NCERT Class 12, Exercise 13.1)

Solution — Step by Step

P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad \text{provided } P(B) \neq 0

This reads: “the probability of $A$ given that $B$ has occurred.”

P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.1}{0.3} = \mathbf{\frac{1}{3}}

If we know event $B$ has happened, the probability that $A$ also happened is $1/3 \approx 0.333$ .

Notice that $P(A|B) = 1/3 < P(A) = 0.4$ . This means knowing $B$ occurred actually decreases the chance of $A$ — the events are negatively correlated.

Why This Works

Conditional probability restricts the sample space. When we know $B$ has occurred, the entire universe shrinks to just $B$ . The only part of $A$ that matters is the part that overlaps with $B$ — which is $A \cap B$ . The ratio $P(A \cap B)/P(B)$ gives us the proportion of $B$ that also belongs to $A$ .

Think of it with a Venn diagram: $B$ is your new “whole world,” and $A \cap B$ is the portion of that world where $A$ is true.

Alternative Method — Check independence and find P(B|A) too

We can also verify: are $A$ and $B$ independent?

For independence, we need $P(A \cap B) = P(A) \cdot P(B) = 0.4 \times 0.3 = 0.12$ .

But $P(A \cap B) = 0.1 \neq 0.12$ . So $A$ and $B$ are not independent.

Also, $P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.1}{0.4} = \frac{1}{4} = 0.25$ .

Note: $P(A|B) \neq P(B|A)$ in general — this is a common source of confusion.

The title mentions Bayes’ theorem, but for this direct calculation, the simple conditional probability formula suffices. Bayes’ theorem is useful when $P(A \cap B)$ is not directly given and you need to compute it from $P(B|A)$ and $P(A)$ : $P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$ .

Common Mistake

The biggest conceptual error: confusing $P(A|B)$ with $P(B|A)$ . Students write $P(A|B) = P(A \cap B)/P(A)$ instead of dividing by $P(B)$ . Remember — the event after the bar ( $B$ ) goes in the denominator. “Given $B$ ” means $B$ is the condition, so $P(B)$ is what you divide by. Writing it wrong gives $0.1/0.4 = 0.25$ — a different answer that looks equally reasonable.

Question

Solution — Step by Step

Why This Works

Alternative Method — Check independence and find P(B|A) too

Common Mistake

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