Q.If $P(A) = \frac{7}{13}$, $P(B) = \frac{9}{13}$ and $P(A \cap B) = \frac{4}{13}$, evaluate $P(A|B)$.

Concept: Conditional Probability — the probability of $A$ given $B$ is the ratio of their intersection to the probability of $B$.

Step 1: Write the formula for conditional probability:

Step 2: Substitute the given values:

Step 3: Simplify by cancelling $\frac{1}{13}$:

Conditional probability answers the question: If we know that event B has occurred, how does that change the chance that event A also occurs? The key insight is that knowing B happened restricts the "sample space" to just the outcomes in B. So instead of measuring $P(A)$ against the whole space, we measure $P(A \cap B)$ — the part of A that lies inside B — against $P(B)$.

This is exactly the formula:

It works because we are renormalising the probability of the overlap by the probability of the new "universe" (B). No extra conditions needed — just plug in the numbers.

1. Identify the given probabilities:

   $P(A) = \frac{7}{13}$, $P(B) = \frac{9}{13}$, $P(A \cap B) = \frac{4}{13}$.

2. Write the definition of conditional probability:

3. Substitute the known values:

4. Simplify the fraction: The $\frac{1}{13}$ cancels in numerator and denominator, leaving

Method: Computing a conditional probability from the three basic quantities

Use this direct approach whenever you are handed $P(A)$, $P(B)$ and $P(A\cap B)$ and asked for a conditional probability.

Steps

Step 1: Identify the conditioning event — it sets the denominator.

The event written after the vertical bar is the one you are "given," so its probability goes in the denominator. For $P(A\mid B)$ the condition is $B$.

Step 2: Apply the definition.

The numerator is always the joint probability $P(A\cap B)$ — the overlap — never $P(A)$ on its own.

Step 3: Substitute and simplify; ignore any unused data.

Put the given fractions in and simplify. Questions often supply an extra value (such as $P(A)$) that is not needed — recognising that it plays no role is part of the skill, not a sign you missed a step.

Common Mistakes

Mistake 1: Putting $P(A)$ in the numerator instead of $P(A\cap B)$.

Why it's wrong: conditional probability measures only the part of $A$ lying inside $B$, which is $P(A\cap B)$. Using $\tfrac{7/13}{9/13}$ gives $\tfrac79$, a wrong answer. Correct approach: always use $P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}=\dfrac{4/13}{9/13}=\dfrac49$.

Mistake 2: Trying to use the unneeded value $P(A)=\tfrac{7}{13}$.

Why it's wrong: the formula needs only $P(A\cap B)$ and $P(B)$; the extra datum is a distractor. Correct approach: recognise which quantities the formula actually requires and ignore the rest.