Mathematics 9709 · Probability & Statistics 1

Multiplication & independence (AND)

AND means both things must happen.
If the events are independent, the second one is not affected by the first.

AND means multiply when nothing changes

For independent events, $P(A\cap B)=P(A)P(B)$ . On a tree: multiply along a path, add different paths that work.

Independent does not mean different. It means the chance of one event stays the same after the other event happens.
A tree is just a tidy way to list paths. Each full path is an AND statement.
A possibility grid does the same job when all pairs of outcomes are equally likely.

Worked example

First rung: one coin and one die

A fair coin is tossed and a fair die is rolled. Find the probability of head and 6.

The coin and die do not affect each other.
Multiply the two chances: $\tfrac12\times\tfrac16=\tfrac1{12}$ .

Answer

\dfrac1{12}

Two dice give 36 equally likely cells. Product even means every cell except odd times odd, so 27 cells win.

To prove independence, compare two numbers.

Find $P(A)$ and $P(B)$ .
Find $P(A\cap B)$ from the data.
Work out $P(A)P(B)$ .
If $P(A\cap B)=P(A)P(B)$ , they are independent. If not, they are not.

Worked example

Independence test from a table

In a college table, $X$ means Year 1 and $N$ means netball. The table gives $P(X\cap N)=\tfrac{12}{120}$ , $P(X)=\tfrac{50}{120}$ , and $P(N)=\tfrac{40}{120}$ . (9709/52 O/N 2024 Q1)

Multiply the separate chances: $P(X)P(N)=\tfrac{50}{120}\times\tfrac{40}{120}=\tfrac{5}{36}$ .
Compare with the real overlap: $P(X\cap N)=\tfrac{12}{120}$ .
They are not equal.

Answer

X\text{ and }N\text{ are not independent}

Common mistake

Mutually exclusive and independent are not the same thing. If two non-impossible events cannot both happen, then knowing one happened changes the chance of the other to zero. That is dependent.

A second way to test independence

If $P(A\mid B)=P(A)$ , the condition did not change the chance of A.
That also means A and B are independent.
In exams, the product test is usually cleaner because it uses the table directly.

PreviousThe addition law (OR)NextConditional probability

Multiplication & independence (AND)

AND means both things must happen.
If the events are independent, the second one is not affected by the first.

AND means multiply when nothing changes

For independent events, $P(A\cap B)=P(A)P(B)$ . On a tree: multiply along a path, add different paths that work.

Independent does not mean different. It means the chance of one event stays the same after the other event happens.
A tree is just a tidy way to list paths. Each full path is an AND statement.
A possibility grid does the same job when all pairs of outcomes are equally likely.

Worked example

First rung: one coin and one die

A fair coin is tossed and a fair die is rolled. Find the probability of head and 6.

The coin and die do not affect each other.
Multiply the two chances: $\tfrac12\times\tfrac16=\tfrac1{12}$ .

Answer

\dfrac1{12}

Two dice give 36 equally likely cells. Product even means every cell except odd times odd, so 27 cells win.

To prove independence, compare two numbers.

Find $P(A)$ and $P(B)$ .
Find $P(A\cap B)$ from the data.
Work out $P(A)P(B)$ .
If $P(A\cap B)=P(A)P(B)$ , they are independent. If not, they are not.

Worked example

Independence test from a table

In a college table, $X$ means Year 1 and $N$ means netball. The table gives $P(X\cap N)=\tfrac{12}{120}$ , $P(X)=\tfrac{50}{120}$ , and $P(N)=\tfrac{40}{120}$ . (9709/52 O/N 2024 Q1)

Multiply the separate chances: $P(X)P(N)=\tfrac{50}{120}\times\tfrac{40}{120}=\tfrac{5}{36}$ .
Compare with the real overlap: $P(X\cap N)=\tfrac{12}{120}$ .
They are not equal.

Answer

X\text{ and }N\text{ are not independent}

Common mistake

Mutually exclusive and independent are not the same thing. If two non-impossible events cannot both happen, then knowing one happened changes the chance of the other to zero. That is dependent.

A second way to test independence

If $P(A\mid B)=P(A)$ , the condition did not change the chance of A.
That also means A and B are independent.
In exams, the product test is usually cleaner because it uses the table directly.