Conditional Probability, Independence, and Trees

Scope Label

Core 9758. This note develops the second layer of H2 probability: how probabilities change after information is given, how multi-stage processes are represented, and what independence means.

Use it with the hub Probability.

Conditional Probability as a Reduced Sample Space

Conditional probability means probability after some information is known.

The probability of $A$ given $B$ is

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

This formula should be read conceptually:

restrict attention to the part of the sample space where $B$ occurred, then ask what proportion also lies in $A$.

Caption: Conditional probability changes the reference universe from the whole sample space to the event that is already known to have occurred.

Example: Die Reduced Sample Space

A fair die is thrown. Let

$$A=\{\text{even number}\},B=\{\text{number greater than }3\}.$$

Then

$$A=\{2,4,6\},B=\{4,5,6\}.$$

Given that $B$ occurred, the relevant sample space is no longer $S=\{1,2,3,4,5,6\}$. It is only

$$B=\{4,5,6\}.$$

Inside $B$, the even outcomes are $\{4,6\}$. Hence

$$P(A\mid B)=\frac{2}{3}.$$

Caption: Conditioning on $B$ means only the outcomes inside $B$ remain relevant.

Why $P(A\mid B)$ and $P(B\mid A)$ Are Different

The expressions $P(A\mid B)$ and $P(B\mid A)$ use the same overlap $A\cap B$, but they divide by different reference spaces:

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

So they are generally not equal.

Caption: $P(A\mid B)$ and $P(B\mid A)$ use the same overlap but different reference spaces.

Bayes’ Theorem

Bayes’ theorem is a way to reverse a conditional probability when the reverse direction is easier to model.

Starting from

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

and also

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

we get

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

This is useful when a question gives information such as a test result and asks for the probability of the underlying cause. The denominator $P(B)$ must refer to the total probability of observing $B$, often found by splitting into cases.

General Multiplication Rule

Starting from

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

we get

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

Equivalently,

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

This is the general multiplication rule. It does not require independence.

Caption: The multiplication rule can be read as the probability of moving along a path into $B$ and then into $A$ within $B$.

Tree Diagrams

Tree diagrams are useful for multi-stage experiments, especially when probabilities change from one stage to the next.

The rule is:

• multiply along a branch to get a path probability
• add the path probabilities that satisfy the required event

For example, if a bag has changing composition after a draw without replacement, a tree diagram records the conditional probabilities on the second draw.

Caption: In a tree diagram, multiply along a branch to get a path probability, then add the relevant path probabilities for the required event.

Independence

Events $A$ and $B$ are independent if knowing that one occurred does not change the probability of the other.

This can be written as

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

provided $P(B)>0$.

Equivalently,

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

This product form is often the easiest way to test independence.

Caption: For independent events, the proportion of $A$ remains the same after restricting the sample space to $B$.

Independence Is Not Mutual Exclusivity

Mutually exclusive events cannot happen together:

$$P(A\cap B)=0.$$

Independent events preserve probabilities:

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

These are different ideas. If $P(A)>0$ and $P(B)>0$, mutually exclusive events cannot be independent, because knowing $B$ occurred makes $A$ impossible.

Caption: Mutually exclusive events concern overlap; independent events concern whether one event changes the probability of the other.

Choosing the Right Representation

Use the representation that matches the structure:

Structure	Representation
Two categorical variables	table
Event overlap	Venn diagram
Stages with changing probabilities	tree diagram
Selection from equally likely outcomes	counting
Given information	conditional probability

Caption: A disciplined probability solution moves from experiment to sample space, event definition, representation, and calculation.

Worked Core Examples

Example 1: Conditional probability

Suppose

$$P(A)=0.5,P(B)=0.6,P(A\cap B)=\mathrm{0.3.}$$

Then

$$P(A\mid B)=\frac{0.3}{0.6}=\mathrm{0.5.}$$

Example 2: Multiplication rule

If

$$P(B)=0.4,P(A\mid B)=0.7,$$

then

$$P(A\cap B)=P(A\mid B)P(B)=0.7(0.4)=\mathrm{0.28.}$$

Example 3: Independence test

Suppose

$$P(A)=0.4,P(B)=0.5,P(A\cap B)=\mathrm{0.2.}$$

Since

$$P(A)P(B)=0.4(0.5)=0.2=P(A\cap B),$$

the events are independent.

Example 4: Tree path probability

A test has two stages. Suppose the probability of passing stage 1 is $0.8$, and the probability of passing stage 2 given stage 1 is passed is $0.7$.

The probability of passing both stages is

$$0.8\times 0.7=\mathrm{0.56.}$$

This is multiplication along a path.

Common Pitfalls

• Treating $P(A\mid B)$ and $P(B\mid A)$ as the same.
• Using $P(A\cap B)=P(A)P(B)$ without independence.
• Forgetting that tree branches after the first stage are conditional probabilities.
• Adding branch probabilities when the paths should be multiplied.
• Confusing mutually exclusive events with independent events.
• Thinking a Venn diagram can visually prove independence without calculation.

Links

• Probability
• Events, Probability Laws, and Complements
• Permutations and Combinations