Topic 2.6 Notes – Conditional Probability

What Conditional Probability Is

A conditional probability is the probability of event $A$ given that event $B$ has happened. It is written $P(A \mid B)$.

A few reminders help this make sense:

• The bar means “given.”
• The event after the bar is the condition.
• $A \cap B$ means both $A$ and $B$ happen.

The whole idea is that the condition changes your sample space. If you know $B$ occurred, then outcomes outside $B$ are no longer possible. You only look inside $B$. In the Venn diagram, that means focusing on circle $B$ and asking what part of it overlaps with $A$.

Conditional probability with a Venn diagram

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

This works as long as $P(B)>0$.

Read it like this:

• numerator $P(A \cap B)$ = probability of being in both events
• denominator $P(B)$ = probability of the condition
• meaning = among the outcomes in $B$, what proportion are also in $A$?

If $P(B)=0$, then $P(A \mid B)$ is undefined because you would be dividing by 0.

How to Find Conditional Probability

Most mistakes come from using the wrong denominator. The condition tells you the denominator.

1. Identify the condition.
   • In $P(A \mid B)$, the condition is $B$.
   • That means your reference group is only outcomes in $B$.
2. Find the overlap.
   • The numerator is $A \cap B$, the outcomes in both events.
3. Divide overlap by condition.

From probabilities,

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

From counts,

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

Suppose a table shows 80 bus riders, and 24 of them were late. Then

$$P(\text{Late} \mid \text{Bus})=\frac{24}{80}=0.30$$

That means among bus riders, 30\% were late.

In a two-way table:

• use the cell for “both” as the numerator
• use the row or column total for the condition as the denominator

Words that usually signal conditional probability:

• given
• among
• of those
• restricted to

The General Multiplication Rule

If you rearrange the conditional probability formula, you get the general multiplication rule:

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

You can also write

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

This is how you find the probability that both events happen when one probability is conditional on the other.

Example:

• $P(A)=0.35$
• $P(B \mid A)=0.60$

Then

$$P(A \cap B)=(0.35)(0.60)=0.21$$

So 21% of the whole population is in both $A$ and $B$.

This rule does not assume independence. That matters a lot on AP Stats.

Tree Diagrams and Sequential Events

A tree diagram helps when events happen in stages, especially when later probabilities change based on earlier outcomes.

• Each branch shows a possible next outcome.
• Branches from the same node add to 1.
• To get the probability of a full path, multiply along the path.

That path product is just the multiplication rule in picture form.

Without replacement is the classic case. If you draw two cards and the first is an ace, then the second-draw probabilities change. This tree shows all four possible two-draw paths and the product for each one.

Two-card tree diagram without replacement

For two aces in a row, follow the Ace → Ace path:

$$P(A_1 \cap A_2)=\frac{4}{52}\cdot\frac{3}{51}=\frac{1}{221}$$

A common AP mistake is putting original probabilities on second-stage branches. Those second branches should usually be conditional probabilities.

What Students Mix Up

• $P(A \mid B)$ and $P(B \mid A)$ use the same overlap but different denominators, so they usually are not equal.
• $P(A \cap B)$ is joint probability. It uses the whole sample space.
• $P(A \mid B)$ is conditional probability. It uses only the condition as the reference group.
• If the problem says “given,” “among,” or “of those,” do not divide by the grand total.
• Do not replace $P(B \mid A)$ with $P(B)$ unless independence is given or proven later.
• “Given” means known information, not necessarily something that happened first.
• Conditional probability does not show causation.