Year 7AC9M7P01

Probability of Single-Step Events

Identify the sample space for single-stage events; assign probabilities to the outcomes of these events and predict relative frequencies for related events.

Probability is a fraction: the outcomes you want over all the outcomes there are. Listing the sample space properly does most of the work.

Method

The sample space is every possible outcome

List it before calculating anything.

The formula

P(E) = \frac{\text{favourable outcomes}}{\text{total outcomes}}

Only valid when all outcomes are equally likely.

Probabilities run from 0 to 1

0 \leq P(E) \leq 1

0 is impossible, 1 is certain. An answer outside this is wrong.

All probabilities add to 1

P(E) + P(\text{not } E) = 1

Useful shortcut when 'not' is easier to count.

Worked example

e.g.

\text{A bag holds 3 red, 4 blue and 5 green counters. Find } P(\text{blue}).

List or count the sample space. $3 + 4 + 5 = 12 \text{ counters}$
Count the favourable outcomes. $4 \text{ blue}$
Write the fraction. $P(\text{blue}) = \frac{4}{12}$
Simplify. $= \frac{1}{3}$
Check it lies between 0 and 1. $0 < \tfrac{1}{3} < 1 \quad\checkmark$

Practice

Count the sample space first.

P(\text{even on a die})

Answer

\frac{3}{6} = \frac{1}{2}

P(\text{a vowel from the letters of } MATHS)

Answer

\frac{1}{5}

\text{From the bag above, } P(\text{not green})

Answer

\frac{7}{12}

\text{If } P(\text{rain}) = 0{,}3, \text{ find } P(\text{no rain}).

Answer

0{,}7

\text{Why can a probability never be } 1{,}4?

Answer

\text{Nothing is more than certain.}