Maths
What is probability?
Quick answer
Probability measures how likely something is to happen, on a scale from 0 (impossible) to 1 (certain). A fair coin landing heads has a probability of 1/2.
In one sentence
Probability is the maths of chance: it gives a number to how likely something is.
Explained
Some things are certain (the sun will rise tomorrow), some are impossible (rolling a 7 on a normal dice), and most are somewhere in between. Probability puts them on a scale from 0 to 1.
- 0 means impossible
- ½ (0.5) means an even chance, as likely as not
- 1 means certain
sun
bag
ball
Probabilities are usually written as fractions, decimals or percentages, so ½ = 0.5 = 50%.
When all the outcomes are equally likely, there is a simple formula:
probability = number of ways it can happen ÷ total number of outcomes
A bag holds 2 green, 3 yellow and 5 purple sweets, 10 in total. Picking without looking, P(yellow) = 3/10. P(not yellow) = 7/10. Notice that 3/10 + 7/10 = 1. Something either happens or it doesn't, so those two probabilities always add up to 1.
Probability tells you what to expect in the long run, not what will happen next. Flip a coin 10 times and you might get 7 heads. Flip it 1,000 times and the share of heads will usually be close to half. And a coin never "owes" you a tails after a run of heads.
When outcomes are not equally likely (will it rain? will this drawing pin land point up?), we estimate the probability from experiments or records. If a pin lands point up 36 times in 100 throws, the estimated probability is 0.36. This is called relative frequency or experimental probability.
Examples
| Event | Probability |
|---|---|
| Rolling an even number on a dice | 3/6 = ½ |
| Picking a heart from 52 playing cards | 13/52 = ¼ |
| Rolling a number greater than 4 | 2/6 = ⅓ |
| Picking a vowel from the letters of MATHS | 1/5 |
| Two coins both landing heads | ½ × ½ = ¼ |
Common mistakes
- Probabilities above 1 or below 0. They can't exist. If you get 6/5, something has gone wrong.
- Thinking every outcome is equally likely. Rolling two dice, a total of 7 (6 ways out of 36) is far more likely than 12 (1 way).
- The gambler's mistake: expecting the "due" result after a streak. Each throw is independent.
Quick quiz
Probability
Check what you know with 5 quick questions.
Question 1 of 5
You roll a fair six-sided dice. What is the probability of rolling a 4?
Practise it
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Browse maths worksheets for data and chance activities.
For grown-ups
Related topics
Fractions
A fraction is part of a whole that has been split into equal parts. In ¾, the bottom number says how many equal parts and the top says how many you have.
Decimals
Decimals are numbers with a decimal point. Digits after the point show tenths, hundredths and so on, so 2.5 means two and five tenths.
Percentages
A percentage is a number out of 100, written with the sign %. So 25% means 25 out of every 100, which is the same as one quarter.
Mean, median and mode
Mean, median and mode are three kinds of average. The mean shares the total out equally, the median is the middle value in order, and the mode is the most common value.
Bar charts
A bar chart shows data using bars of different lengths. The longer the bar, the bigger the number. In the US it is usually called a bar graph.
Algebra
Algebra is maths that uses letters, like x or n, to stand for numbers we don't know yet or numbers that can change. For example, if n + 3 = 10, then n = 7.