Independent and Conditional Probability

1. In probability, we often perform an experiment (rolling a die, drawing a ticket, checking whether a train is late).
2. An event is a set of outcomes we care about, such as:
   1. “rolling a 6”
   2. “the train is late”
   3. “the student chooses trampolining”
3. In this chapter we focus on:
   1. Conditional probability – probabilities given that we already know something happened.
   2. Independent events – situations where this extra information does not change the probability.
4. These two ideas are closely linked.

Conditional Probability: “Given That”

When you read “given that”, you should think:

The sample space has been restricted to the outcomes where $A$ happens.

So, outside $A$ we no longer care about any outcomes. Inside $A$ we ask: what fraction also lie in $B$?

Conditional Probability Formula

Using this “reduced sample space” idea, we get the rule

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

1. The denominator is $P(A)$ because we are working inside $A$.
2. The numerator is $P(A \cap B)$: the overlap of $A$ and $B$.

Conditional Probability with Venn Diagrams and Tables

Venn Diagram View

1. In a Venn diagram:
   1. The "new sample space" for $P(B \mid A)$ is the region $A$.
   2. The favourable outcomes are in the overlap $A \cap B$.
2. So, using counts, $$P(B \mid A)=\frac{n(A \cap B)}{n(A)} .$$
3. The probability version is the same idea with probabilities instead of counts.

Two-Way (Contingency) Tables

For real data, probabilities often come from a two-way table (for example, gender vs chosen sport).

To find something like $P \text { (Table tennis } \mid \text { Female) }$:

1. Identify the condition (“Female”) → use the female row total as the denominator.