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.
2. Use the intersection cell (“Female and Table tennis”) as the numerator.
3. Form the fraction and simplify.

This method is the same as the Venn diagram approach, just organised in a table.

Joint Probability and Tree Diagrams

Multiplication Rule (General Form)

1. Rearranging the conditional probability formula gives $$P(A \cap B)=P(A) P(B \mid A)$$
2. This is the joint probability rule: to find the probability that both $A$ and $B$ happen, multiply:
   1. the probability of the first event $A$, and
   2. the conditional probability of the second given the first, $P(B \mid A)$.

Tree Diagrams

A tree diagram organises multi-stage experiments (for example, “train late?” then “miss appointment?”).

1. Rules:
   1. Multiply along a branch to get a joint probability, e.g. $P(A \cap B)=P(A) P(B \mid A)$.
   2. Add along different branches (mutually exclusive paths) to get the probability of an event.
2. Tree diagrams are especially helpful when probabilities change after the first event, such as:
   1. problems without replacement
   2. “if it rains, then …” style problems
3. Each second-stage probability is then a conditional probability.

Independent Events

Independence is a statement about information: if event $A$ occurs (or does not occur), it does not give you any extra help in predicting event $B$.

Mutual Exclusivity Is Different From Independence

Students often confuse independent with mutually exclusive.

Three Equivalent Ways To Test Independence

1. There are several equivalent ways to express independence.
2. In practice, you choose the one that matches the information you are given.

Method 1: Multiplication Rule

1. If $A$ and $B$ are independent, then $$P(A\cap B)=P(A)\times P(B)$$
2. This is useful when you can find $P(A\cap B)$ directly (for example from a table, Venn diagram, or tree diagram).

Method 2: Conditional Probability Form

1. If $A$ and $B$ are independent, then $$P(B\mid A)=P(B\mid A')=P(B)$$ where $A'$ means "not $A$".
2. This matches the meaning of independence: the probability of $B$ is unchanged even after learning whether $A$ happened.

Method 3: Conditional Probability Formula

1. Conditional probability is defined by $$P(B\mid A)=\frac{P(A\cap B)}{P(A)} \quad (P(A)>0)$$
2. If events are independent, then $P(A\cap B)=P(A)P(B)$, and substituting into the formula gives $P(B\mid A)=P(B)$.

Independence In Two-Way Tables (Reasoning With Data)

Independence also appears when working with data tables.

Common Patterns: When Independence Is Plausible (And When It Is Not)

1. Independence is not something you assume automatically, you justify it.
2. Events are often independent when:
   1. They come from separate experiments (coin flip and die roll).
   2. There is replacement (drawing a card, replacing it, then drawing again).
   3. The process is designed so one outcome cannot influence the next.
3. Events are often dependent when:
   1. You sample without replacement (taking two doughnuts from a bag and eating the first).
   2. One event is a cause or strong influence of the other (rain affects playing tennis).
   3. The information about one event changes the sample space for the other.