23.1 The probability scale and equally likely outcomes — Learn

Prerequisites: fractions, equivalent fractions and simple ratios.

Objectives: describe chance on a scale from impossible to certain; calculate probabilities when outcomes are equally likely; use complements.

A probability is a number from 0 to 1. Zero means impossible, and 1 means certain. A probability of one half means the event and its opposite are equally likely. Probabilities may be written as fractions, decimals or percentages.

For equally likely outcomes, count the outcomes that match the event and divide by the total number of outcomes:

P(\text{event})=\frac{\text{number of matching outcomes}}{\text{total number of outcomes}}

The complement is the event that does not happen. Its probability is $1-P(\text{event})$ . An event and its complement are mutually exclusive and include every possible outcome, so their probabilities add to 1.

Example 1. A fair six-sided die is rolled. Find the probability of rolling a number greater than 4.

The outcomes are 1, 2, 3, 4, 5 and 6, each equally likely.

Two outcomes, 5 and 6, are greater than 4.

P(\text{greater than 4})=\frac{2}{6}=\frac13

If the die is rolled 12 times, the expected number of results greater than 4 is $12\times\frac13=4$ . The observed number may differ from this expectation.

Example 2. A bag contains 4 green counters and 1 yellow counter. Find the probability of not choosing green.

There are 5 counters, so the probability of green is $\frac45$ .

P(\text{not green})=1-\frac45=\frac15

Watch out. Do not count colours as equally likely when there are different numbers of counters. One yellow counter among five gives probability $\frac15$ , not $\frac12$ .

Investigate the spinner

Predict: if 4 of 10 equal sectors are red, predict the probability of red. Then predict how the probability changes as the number of red sectors increases.

Test: move the control to test your predictions at 0, 4 and 10 red sectors.

Explain: why does the probability reach 0 and 1 at the ends of the scale? Write your explanation before reading on.

Progress check

The spinner has 10 equal sectors. Suppose 4 of them are red. Write the probability of landing on red, and the probability of not landing on red. Then state how many red sectors would make red impossible.

You should now be able to

describe probabilities using the scale from 0 to 1;
calculate probabilities from equally likely outcomes;
relate theoretical probability to an expected number of outcomes;
use the complement to find a probability.