The probability of an event *E* happening, *P(E)*, is given by the number of ways the event can occur, *n(E)*, compared with the total number of outcomes possible *n(S)* (the size of the sample space).

If *P(E)* = 0 the event is impossible, and if *P(E)* = 1 the event is certain.

The sum of all (mutually exclusive) probabilities is 1.

### Examples

1. A container holds 8 blue, 7 white and 10 yellow marbles. If one marble is selected at random, find the probability of getting

a) a white marble

b) a white or blue marble

c) a yellow, white or blue marble

d) a red marble

### Show Answer

The size of the sample space, or total number of marbles is 8 + 7 + 10 = 25.

a)

b)

c)

d) chance - impossible.

2. The probability that a traffic light will turn green as a car approaches it is estimated to be . A taxi goes through 288 intersections where there are traffic lights. How many of these would be expected to turn green as the taxi approached?

### Show Answer

As the expected chance is only , then the theoretical solution is:

times.

## Complementary Events

The complement of an event happening is the event not happening. That is , the complement of *P(E)* is *P(not E)*. We can write this as .

For example:

A die is thrown. Find the probability of:

a) throwing a 4.

b) not throwing a 4.

### Show Answer

a)

b)

In general,

## Non-Mutually Exclusive Events

Sometimes, there is an overlap where more than one event can occur at the same time. We call these non-mutually exclusive events. It is important to count the possible outcomes carefully when this happens - and Venn diagrams help.

For example:

One card is drawn from a set of cards numbered 1 to 12. Find the probability of drawing out an odd number **OR** a multiple of 3.

The odd cards are 1, 3, 5, 7, 9, 11.

The multiples of 3 are 3, 6, 9 and 12.

The numbers 3 and 9 are both odd and multiples of 3 (- try not to count them twice).

So there are 8 numbers that are odd **OR** multiples of 3: 1, * 3*, 5, 6, 7,

*, 11, 12.*

**9***P*(odd or multiples of 3) =

We can also display this in a Venn Diagram:

In fact, there is a formula that can be used for non-mutually exclusive events.

From 50 cards numbered from 1 to 50, one is selected at random. Find the probability that the card selected is even **or** less than 15.

Some cards are both even and less than 15 = 2, 4, 6, 8, 10, 12, 14

*P*(even and < 15) =

*P*(even) =

*P*(< 15) =

*P*(even or < 15) = *P*(even) + *P*(< 15) - *P*(even and < 15)

*P*(even or < 15) =

*P*(even or < 15) =