Topic 4.5 Notes – Conditional Probability

1. Conditional Probability Explained

Suppose you’re looking at students in a school. Let

• A = student is on the soccer team
• B = student is a senior

If you want the probability a student is on the soccer team given that the student is a senior, you write:

$$P(A \mid B)$$

Read it as “the probability of A given B.”

The Formula

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)} \quad \text{as long as } P(B) \ne 0$$

Break that down:

• $A \cap B$ means both A and B happen.
• You divide by $P(B)$ because you are restricting your attention to only the outcomes where B occurred.

That’s the key idea.
When you condition on B, you shrink the sample space to just B. Then you ask: What fraction of that group also satisfies A?

Trigger word on tests: “given.” If you see it, think conditional probability immediately.

2. The Multiplication Rule and Joint Probability

Rearrange the conditional probability formula:

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

This is the general multiplication rule.

You can also write:

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

Both are correct.

Interpretation:

• First event happens.
• Multiply by the probability the second event happens given the first already occurred.

This works for any two events. Independence is not required.

Big connection to remember:

• Conditional probability → divide
• Multiplication rule → multiply

They’re the same relationship, just rearranged.

On multiple-choice questions, they often give you $P(A)$ and $P(B|A)$ and ask for $P(A \cap B)$. That’s a straight multiplication rule situation.

3. Finding Conditional Probability from Different Representations

The structure is always the same:
numerator = joint, denominator = conditioning event.

What changes is how the information is presented.

a. Two-Way Tables

Here’s a typical layout:

Two-way table of instrument participation by class year

Suppose you want $P(\text{Plays} \mid \text{Senior})$.

• Conditioning event = Senior → denominator = 60 (the Senior row total)
• Joint count (Plays AND Senior) = 40 (the cell where Senior and Plays intersect)

$$P(\text{Plays} \mid \text{Senior}) = \frac{40}{60}$$

Common mistake: dividing by 140 (the grand total). Once you see “given Senior,” the total seniors becomes your whole world.

b. Venn Diagrams

Venn diagram with sample space S and events A and B

In a Venn diagram:

• The overlap is $A \cap B$.
• To find $P(A \mid B)$, mentally zoom into circle B.
• Ask: what fraction of circle B is also in A?

You’re comparing the overlap to the entire B circle.

c. Tree Diagrams

Probability tree with conditional branches

Tree diagrams are great for multi-stage processes.

• First branches = initial probabilities.
• Second branches = conditional probabilities.
• Multiply along a path to get a joint probability.

Example from the tree above:

$$P(A \cap B) = 0.6 \times 0.5 = 0.30$$

Each full path represents an intersection.

4. How to Approach Conditional Probability Problems

When you’re stuck, slow it down:

1. Rewrite the question using $P(\_\mid\_)$ notation.
2. Identify the conditioning event → that’s your denominator.
3. Find the joint probability → numerator.
4. Divide.

If you’re given a conditional probability and a marginal probability and need a joint probability, switch to the multiplication rule and multiply instead.

Always sanity-check:

• Answer must be between 0 and 1.
• If you’re conditioning on a small group, the denominator should reflect that smaller group.

5. Conditional Probability and Independence

Two events A and B are independent if learning that B happened does not change the probability of A.

Mathematically, independence means:

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

All three statements say the same thing in different forms.

If the conditional probability changes from the original probability, the events are not independent.

Students often assume independence automatically. On a test, you must either be told the events are independent or verify it using the probabilities.