How Do You Calculate Conditional Probability P(A|B)?

You calculate conditional probability using

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}, \qquad P(B)>0.$$

This means: once event $B$ is known to have occurred, calculate the proportion of outcomes in $B$ that also belong to $A$.

How the Formula Works

The condition $B$ becomes the new, restricted sample space. The numerator counts outcomes where both events occur, while the denominator counts all outcomes satisfying the condition.

This restriction is the key idea: outcomes outside $B$ are ignored, even if they belong to $A$. When using counts rather than probabilities, the same relationship is $\frac{n(A\cap B)}{n(B)}$.

Rearranging the formula also gives the useful multiplication rule:

$$P(A\cap B)=P(B)P(A\mid B).$$

Conditional probability can also be read from a tree diagram: follow the branch representing $B$, then use the probability of $A$ along that restricted branch.

Worked Example

In a group of 30 students, 18 study biology. Of those 18 students, 12 also study chemistry. Let $A$ be the event that a student studies chemistry and $B$ the event that the student studies biology.

Because the student is already known to study biology, only the 18 biology students form the relevant sample space:

$$P(A\mid B)=\frac{12}{18}=\frac{2}{3}\approx 0.667.$$

Therefore, the probability that a randomly selected student studies chemistry, given that they study biology, is $0.667$, or approximately $66.7\%$.

As a quick check, conditional probabilities must lie between $0$ and $1$. If $A$ and $B$ are independent, then $P(A\mid B)=P(A)$; otherwise conditioning changes the probability.

A common misconception is assuming that $P(A\mid B)=P(B\mid A)$. These probabilities usually differ because their denominators represent different restricted sample spaces.

IB Exam Technique

This is part of SL 4.6: Venn diagrams, including combined events and conditional probability. For a “calculate” question, show the substitution into $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$, then give the numerical answer. In a table, tree diagram, or Venn diagram, identify the condition first and use its total as the denominator.