Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. We write the probability of event $B$ occurring given that event $A$ has occurred as $P(B|A)$. The vertical line $|$ is read as "given".

The Conditional Probability Formula

To calculate conditional probability, we use the following formula:

$P(B|A) = \frac{P(A \cap B)}{P(A)}$

Where:

• $P(B|A)$ is the probability of $B$ given $A$.
• $P(A \cap B)$ is the probability of both $A$ and $B$ occurring (the intersection).
• $P(A)$ is the probability of event $A$ occurring (where $P(A) > 0$).

You can also rearrange this formula to find the probability of both events happening:

$P(A \cap B) = P(A) \cdot P(B|A)$

Example: If $P(A) = 0.6$ and $P(B|A) = 0.3$, what is $P(A \cap B)$?

Using the rearranged formula: $P(A \cap B) = 0.6 \cdot 0.3 = 0.18$

Using Two-Way Frequency Tables

Two-way tables make finding conditional probability easy because they organize data into categories. When you see a "given" condition, you simply restrict your total sample space to just that specific row or column.

Example: Imagine a survey of 200 students about whether they studied and whether they passed a test. You are told:

• 100 students studied and passed.
• 20 students studied and failed.
• The total number of students who studied is 120.

Find $P(\text{pass} | \text{studied})$.

Because we are given that the student studied, we only look at the 120 students who studied. We completely ignore the rest of the 200 students. Out of those 120 students, 100 passed.

$P(\text{pass} | \text{studied}) = \frac{100}{120} = \frac{5}{6} \approx 0.833$