Unit E: Statistics and Probability

Chapter 2: Probability

Odds Against

The odds against an event are the odds of an event not occurring. The formula for odds against is

odds against = number of unfavourable outcomes : number of favourable outcomes

Example 4

Five plastic ducks are placed in a bag.

What are the odds of randomly choosing a yellow duck from the bag?
What are the odds against choosing a yellow duck?

Solutions

favourable outcomes (choosing a yellow duck)	4
unfavourable outcomes (not choosing a yellow duck)	1

\begin{array}{rcl} odds in favour & = & number of favourable outcomes : number of unfavourable outcomes \\ = & number of yellow ducks : number ducks that are not yellow \\ = & 4 : 1 \end{array}

The odds of choosing a yellow duck are 4:1.

favourable outcomes (choosing a yellow duck)	4
unfavourable outcomes (not choosing a yellow duck)	1

\begin{array}{rcl} odds against & = & number of unfavourable outcomes : number of favourable outcomes \\ = & number ducks that are not yellow : number of yellow ducks \\ = & 1 : 4 \end{array}

The odds against choosing a yellow duck are 1:4.

Notice that the odds in favour of an event and the odds against the same event are ratios with the same terms in opposite order. The odds for choosing a yellow duck are 4:1 whereas the odds against choosing a yellow duck are 1:4.

If the odds in favour of an event are known, the odds against it can be found using this information.

Example 5

A spinner is divided into eight sections.

What are the odds in favour of the spinner landing on red?
What are the odds against the spinner landing on red?

Solutions

favourable outcomes (landing on red)	3
unfavourable outcomes (not landing on red)	5

\begin{array}{rcl} odds in favour & = & number of favourable outcomes : number of unfavourable outcomes \\ = & number of red sections : number sections that are not red \\ = & 3 : 5 \end{array}

The odds of the spinner landing on red are 3:5.

favourable outcomes (landing on red)	3
unfavourable outcomes (not landing on red)	5

\begin{array}{rcl} odds against & = & number of unfavourable outcomes : number of favourable outcomes \\ = & number sections that are not red : number of red sections \\ = & 5 : 3 \end{array}

The odds against the spinner landing on red are 5:3.

Example 6

A bag contains 3 yellow jelly beans, 4 green jelly beans, 5 red jelly beans, and 1 black jelly bean.

What is the probability of choosing a green jelly bean, stated as a fraction?
What are the odds in favour of choosing a green jelly bean?
What are the odds against choosing a green jelly bean?
What are the odds of choosing a red or black jelly bean?

Solutions

$\begin{array}{rcl} probability & = & \frac{number of favourable outcomes}{total number of possible outcomes} \\ = & \frac{number of green jelly beans}{total number of jelly beans} \\ = & \frac{4}{13} \end{array}$

favourable outcomes (green jelly bean)	4
unfavourable outcomes (not a green jelly bean)	9

\begin{array}{rcl} odds in favour & = & number of favourable outcomes : number of unfavourable outcomes \\ = & number of green jelly beans : number of jelly beans that are not green \\ = & 4 : 9 \end{array}

The odds in favour of choosing a green jelly bean are 4:9.

favourable outcomes (a green jelly bean)	4
unfavourable outcomes (not a green jelly bean)	9

\begin{array}{rcl} odds against & = & number of unfavourable outcomes : number of favourable outcomes \\ = & number of jelly beans that are not green : number of green jelly beans \\ = & 9 : 4 \end{array}

The odds against choosing a green jelly bean are 9:4.

favourable outcomes (red or black jelly bean)	5 + 1 = 6
unfavourable outcomes (not a red or black jelly bean)	7

Recall: The word or means the two values need to be added.

\begin{array}{rcl} odds in favour & = & number of favourable outcomes : number of unfavourable outcomes \\ = & number of red or black jelly beans : number of jelly beans that are not red or black \\ = & 6 : 7 \end{array}

The odds of choosing a red or black jelly bean are 6:7.