Mathematics 9709 · Probability & Statistics 1

The addition law (OR)

OR means the event is allowed to happen in either way.
If the two ways overlap, the middle gets counted twice unless you subtract it once.

OR means add, then fix the overlap

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$ . If A and B cannot both happen, then $P(A\cap B)=0$ .

Worked example

First rung: no overlap

A fair die is rolled. Find the probability of getting a 1 or a 6.

Getting 1 and getting 6 cannot happen together.
Add the two separate chances: $\tfrac16+\tfrac16=\tfrac26$ .

Answer

\dfrac13

Mutually exclusive means no shared outcomes. The overlap is empty.
Neither means outside both circles: $1-P(A\cup B)$ .
Exactly one means A only or B only. Do not include the overlap.

Worked example

One extra step: fix the overlap

In a group, $P(A)=0.50$ , $P(B)=0.40$ , and $P(A\cap B)=0.10$ . Find $P(A\cup B)$ .

Add A and B, but the overlap has been counted twice.
Subtract the overlap once: $0.50+0.40-0.10=0.80$ .

Answer

0.80

Fill the overlap first. Then fill the two only-regions. The outside is what is left to make 1.

Worked example

Venn diagram numbers

50% own a desktop, 60% own a laptop, and 15% own both. Find the probability that a person owns neither.

Desktop only: $0.50-0.15=0.35$ .
Laptop only: $0.60-0.15=0.45$ .
Inside the circles: $0.35+0.15+0.45=0.95$ .
Outside both: $1-0.95=0.05$ .

Answer

P(\text{neither})=0.05

Words in the question	What to do
A or B	$P(A)+P(B)-P(A\cap B)$
A or B, mutually exclusive	$P(A)+P(B)$
neither A nor B	$1-P(A\cup B)$
exactly one of A and B	add the two only-regions

Common mistake

The total written inside a circle is not the same as the only-region. If

P(A)=0.50

and the middle is

0.15

, then A only is

0.35

, not

0.50

Three-circle Venn diagrams

Put the centre in first.
Fill each pairwise overlap after subtracting the centre.
Fill each only-region last.
Then check everything adds to the total.

PreviousProbability basics NextMultiplication & independence (AND)

The addition law (OR)

OR means the event is allowed to happen in either way.
If the two ways overlap, the middle gets counted twice unless you subtract it once.

OR means add, then fix the overlap

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$ . If A and B cannot both happen, then $P(A\cap B)=0$ .

Worked example

First rung: no overlap

A fair die is rolled. Find the probability of getting a 1 or a 6.

Getting 1 and getting 6 cannot happen together.
Add the two separate chances: $\tfrac16+\tfrac16=\tfrac26$ .

Answer

\dfrac13

Mutually exclusive means no shared outcomes. The overlap is empty.
Neither means outside both circles: $1-P(A\cup B)$ .
Exactly one means A only or B only. Do not include the overlap.

Worked example

One extra step: fix the overlap

In a group, $P(A)=0.50$ , $P(B)=0.40$ , and $P(A\cap B)=0.10$ . Find $P(A\cup B)$ .

Add A and B, but the overlap has been counted twice.
Subtract the overlap once: $0.50+0.40-0.10=0.80$ .

Answer

0.80

Fill the overlap first. Then fill the two only-regions. The outside is what is left to make 1.

Worked example

Venn diagram numbers

50% own a desktop, 60% own a laptop, and 15% own both. Find the probability that a person owns neither.

Desktop only: $0.50-0.15=0.35$ .
Laptop only: $0.60-0.15=0.45$ .
Inside the circles: $0.35+0.15+0.45=0.95$ .
Outside both: $1-0.95=0.05$ .

Answer

P(\text{neither})=0.05

Words in the question	What to do
A or B	$P(A)+P(B)-P(A\cap B)$
A or B, mutually exclusive	$P(A)+P(B)$
neither A nor B	$1-P(A\cup B)$
exactly one of A and B	add the two only-regions

Common mistake

The total written inside a circle is not the same as the only-region. If

P(A)=0.50

and the middle is

0.15

, then A only is

0.35

, not

0.50

Three-circle Venn diagrams

Put the centre in first.
Fill each pairwise overlap after subtracting the centre.
Fill each only-region last.
Then check everything adds to the total.