AND and independent events | CAIE 9709 Probability & Statistics 1 Notes

Chapters

Probability & Statistics 1

Mathematics9709

Independent means the chance is unchanged

Events A and B are independent when knowing one happened does not change the probability of the other.
For independent events, $P(A\cap B)=P(A)P(B)$ .
Separate random devices, replacement and repeated fair trials are common clues, but use the wording or an exact test before declaring independence.

Worked example

A coin and a die

A fair coin and fair die do not affect each other.

Answer

P(H\text{ and }6)=\frac1{12}

Worked example

At least one six in two independent rolls

“At least one six” has several routes. Its complement, no six on either roll, has one route.

P(\text{no six})=\frac56\times\frac56=\frac{25}{36}

Answer

P(\text{at least one six})=\frac{11}{36}

A grid counts complete pairs

Swipe left or right to see the whole diagram →

The 36 ordered pairs are equally likely. An even product fails only when both dice are odd, so 27 of the 36 cells succeed.

An even product occurs unless both dice are odd. There are $3\times3=9$ odd-odd cells among 36 ordered pairs, leaving 27 successful cells.

Answer

P(\text{even product})=\frac{27}{36}=\frac34

In a grid, each cell is an AND route. Count cells only when all cells represent equally likely ordered outcomes.

Prove independence by comparing two values

Find the actual overlap. Calculate $P(A\cap B)$ from the table, diagram or stated data.
Find the product. Calculate $P(A)P(B)$ separately.
Compare exactly. Equality proves independence. Inequality proves dependence.

Worked example

A table does not support independence

A table gives $P(X\cap N)=12/120$ , $P(X)=50/120$ and $P(N)=40/120$ . (9709/52/O/N/24 Q1)

$P(X)P(N)=\frac{50}{120}\times\frac{40}{120}=\frac5{36}$ , which is not $12/120$ .

Answer

X and N are not independent.

Common mistake

Exclusive and independent are not synonyms. If two non-impossible events cannot occur together, then observing one makes the probability of the other zero. They are dependent.

Your turnIndependent check [5 marks]

$P(A)=0.4$ , $P(B)=0.3$ and $P(A\cup B)=0.58$ . Find $P(A\cap B)$ and determine whether A and B are independent.

Show worked answer

The addition rule gives $P(A\cap B)=0.4+0.3-0.58=0.12$ .

Also $P(A)P(B)=0.4(0.3)=0.12$ .

Answer

A and B are independent.

Independent means the chance is unchanged

Events A and B are independent when knowing one happened does not change the probability of the other.
For independent events, $P(A\cap B)=P(A)P(B)$ .
Separate random devices, replacement and repeated fair trials are common clues, but use the wording or an exact test before declaring independence.

Worked example

A coin and a die

A fair coin and fair die do not affect each other.

Answer

P(H\text{ and }6)=\frac1{12}

Worked example

At least one six in two independent rolls

“At least one six” has several routes. Its complement, no six on either roll, has one route.

P(\text{no six})=\frac56\times\frac56=\frac{25}{36}

Answer

P(\text{at least one six})=\frac{11}{36}

A grid counts complete pairs

Swipe left or right to see the whole diagram →

The 36 ordered pairs are equally likely. An even product fails only when both dice are odd, so 27 of the 36 cells succeed.

An even product occurs unless both dice are odd. There are $3\times3=9$ odd-odd cells among 36 ordered pairs, leaving 27 successful cells.

Answer

P(\text{even product})=\frac{27}{36}=\frac34

In a grid, each cell is an AND route. Count cells only when all cells represent equally likely ordered outcomes.

Prove independence by comparing two values

Find the actual overlap. Calculate $P(A\cap B)$ from the table, diagram or stated data.
Find the product. Calculate $P(A)P(B)$ separately.
Compare exactly. Equality proves independence. Inequality proves dependence.

Worked example

A table does not support independence

A table gives $P(X\cap N)=12/120$ , $P(X)=50/120$ and $P(N)=40/120$ . (9709/52/O/N/24 Q1)

$P(X)P(N)=\frac{50}{120}\times\frac{40}{120}=\frac5{36}$ , which is not $12/120$ .

Answer

X and N are not independent.

Common mistake

Exclusive and independent are not synonyms. If two non-impossible events cannot occur together, then observing one makes the probability of the other zero. They are dependent.

Your turnIndependent check [5 marks]

$P(A)=0.4$ , $P(B)=0.3$ and $P(A\cup B)=0.58$ . Find $P(A\cap B)$ and determine whether A and B are independent.

Show worked answer

The addition rule gives $P(A\cap B)=0.4+0.3-0.58=0.12$ .

Also $P(A)P(B)=0.4(0.3)=0.12$ .

Answer

A and B are independent.