Probability
Probability is about estimating the chance or how likely (probable) something is to happen. For example, an outcome when throwing a die or tossing a coin.
We often use words to describe how probable we think it is that an event will take place.
For example, we might say that it is likely to be sunny tomorrow, or that it is unlikely to snow in August.
Have a look at the statements below, and decide whether the following events are:
- certain
- likely
- even (neither likely or unlikely)
- unlikely
- impossible
Question
- You buy a lottery ticket and win the jackpot
- You toss a coin and get heads
- Christmas will fall on 25 December this year
- You grow another nose
- It will rain in the first week of December
- You buy a lottery ticket and win the jackpot - Unlikely
- You toss a coin and get heads - Even chance
- Christmas will fall on 25 December this year - Certain
- You grow another nose - Impossible
- It will rain in the first week of December - Likely
Words like 'certain', 'likely', and so on, may not mean the same to everyone.
We need to be more precise about how likely something (an outcome) is to happen.
The probability of an outcome can have any value between 0 (impossible) and 1 (certain).
It may be written as a fraction, decimal fraction or percentage.
We can calculate a value for the probability using the formula:
\(probability\,of\,an\,outcome = \frac{{number\,of\,ways\,the\,outcome\,can\,happen}}{{total\,number\,of\,possible\,outcomes}}\)
The table below gives some examples of events and how their probability can be calculated.
| Event | Outcome | Number of ways to get this outcome | Total number of possible outcomes | Probability of outcome |
| Throwing a fair, 6-sided dice | Getting an odd number | 3 | 6 | \(\frac{3}{6} = \frac{1}{2}\) |
| Throwing a fair coin | Getting 'tails' | 1 | 2 | \(50\%\) |
| Choosing a playing card from a full pack without looking | The suit being spades | 13 | 52 | \(\frac{13}{52} = \frac{1}{4}\) |
| Choosing a playing card from a full pack without looking | The card being a 'ten' | 4 | 52 | \(\frac{4}{52} = \frac{1}{13}\) |
| Throwing a fair, 6-sided dice | Getting a number less than 5 | 4 | 6 | \(\frac{4}{6} = \frac{2}{3}\) |
| Event | Throwing a fair, 6-sided dice |
|---|---|
| Outcome | Getting an odd number |
| Number of ways to get this outcome | 3 |
| Total number of possible outcomes | 6 |
| Probability of outcome | \(\frac{3}{6} = \frac{1}{2}\) |
| Event | Throwing a fair coin |
|---|---|
| Outcome | Getting 'tails' |
| Number of ways to get this outcome | 1 |
| Total number of possible outcomes | 2 |
| Probability of outcome | \(50\%\) |
| Event | Choosing a playing card from a full pack without looking |
|---|---|
| Outcome | The suit being spades |
| Number of ways to get this outcome | 13 |
| Total number of possible outcomes | 52 |
| Probability of outcome | \(\frac{13}{52} = \frac{1}{4}\) |
| Event | Choosing a playing card from a full pack without looking |
|---|---|
| Outcome | The card being a 'ten' |
| Number of ways to get this outcome | 4 |
| Total number of possible outcomes | 52 |
| Probability of outcome | \(\frac{4}{52} = \frac{1}{13}\) |
| Event | Throwing a fair, 6-sided dice |
|---|---|
| Outcome | Getting a number less than 5 |
| Number of ways to get this outcome | 4 |
| Total number of possible outcomes | 6 |
| Probability of outcome | \(\frac{4}{6} = \frac{2}{3}\) |
Example
Imagine there are two fair spinners.
A triangular one is marked with one number on each side: 1, 2 and 3.
A square one is also marked with one number on each side: 0, 1, 2 and 3.
The total score is the sum of the two numbers the spinners land on.
Jot down, systematically, all the possible outcomes for the two spinners. You will find it useful to use a table of results, as shown.
| Triangular spinner | Square spinner | Total score |
| 1 | 0 | 1 |
| 1 | 1 | 2 |
| 1 | 2 | 3 |
| 1 | 3 | 4 |
| 2 | 0 | 2 |
| 2 | 1 | 3 |
| 2 | 2 | 4 |
| 2 | 3 | 5 |
| 3 | 0 | 3 |
| 3 | 1 | 4 |
| 3 | 2 | 5 |
| 3 | 3 | 6 |
| Triangular spinner | 1 |
|---|---|
| Square spinner | 0 |
| Total score | 1 |
| Triangular spinner | 1 |
|---|---|
| Square spinner | 1 |
| Total score | 2 |
| Triangular spinner | 1 |
|---|---|
| Square spinner | 2 |
| Total score | 3 |
| Triangular spinner | 1 |
|---|---|
| Square spinner | 3 |
| Total score | 4 |
| Triangular spinner | 2 |
|---|---|
| Square spinner | 0 |
| Total score | 2 |
| Triangular spinner | 2 |
|---|---|
| Square spinner | 1 |
| Total score | 3 |
| Triangular spinner | 2 |
|---|---|
| Square spinner | 2 |
| Total score | 4 |
| Triangular spinner | 2 |
|---|---|
| Square spinner | 3 |
| Total score | 5 |
| Triangular spinner | 3 |
|---|---|
| Square spinner | 0 |
| Total score | 3 |
| Triangular spinner | 3 |
|---|---|
| Square spinner | 1 |
| Total score | 4 |
| Triangular spinner | 3 |
|---|---|
| Square spinner | 2 |
| Total score | 5 |
| Triangular spinner | 3 |
|---|---|
| Square spinner | 3 |
| Total score | 6 |
Use the table to answer these questions.
Question
How many different possible outcomes are there?
There are 12 different possible outcomes.
All you have to do is count the number of rows in the table.
Question
How many outcomes give a total score of 2?
Only 2 outcomes give a total score of 2.
These are 1 + 1 and 2 + 0
Question
What is the probability of getting a total score of 2?
The number of outcomes giving a total score of 2 over the number of outcomes possible \(= \frac{2}{{12}}\,or\,\frac{1}{6}\)
Question
How many outcomes give a total score of 4?
3 outcomes give a total score of 4.
Question
What is the probability of getting a total score of 4?
The number of outcomes giving a total score of 4 over the number of outcomes possible \(= \frac{3}{{12}}\,or\,\frac{1}{4}\)
