Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). For equally likely outcomes, P(event) = number of favourable outcomes ÷ total number of outcomes. The probabilities of all the possible outcomes add up to 1, so P(not A) = 1 − P(A). Probabilities are written as fractions, decimals or percentages, never as ratios.
Probability basics
- 0 ≤ P(event) ≤ 1
- P(event) = favourable outcomes ÷ total outcomes (equally likely outcomes)
- All outcomes add up to 1
- P(not A) = 1 − P(A)
The probability scale
Every probability lies between 0 and 1. An impossible event has probability 0, a certain event has probability 1, and an even chance is ½ or 0.5. Words such as “unlikely” and “likely” describe positions on the scale.
Equally likely outcomes
When every outcome is equally likely, the probability of an event is the number of ways it can happen divided by the total number of outcomes. A bag of 4 red, 7 blue and 9 green counters has 20 counters, so P(blue) = 7/20. Simplify fractions where possible.
Probabilities add up to 1
The probabilities of all the possible outcomes of an experiment add up to 1. This lets you find a missing probability: if a spinner has P(A) = 0.1 and P(B) = 0.36, and C and D are the only other outcomes, then P(C) + P(D) = 0.54. It also gives the rule P(not A) = 1 − P(A).
Writing probabilities
Probabilities can be fractions, decimals or percentages, such as 1/4, 0.25 or 25%. Do not write them as ratios, such as 1 : 4, or in words like “1 in 4” in an exam answer, because these can lose marks.
Theoretical and experimental probability
Probabilities found by counting equally likely outcomes are theoretical. When outcomes are not equally likely, such as a biased spinner or whether it rains, probabilities are estimated from experiments or past data instead. Both kinds follow the same rules: they lie between 0 and 1, and all the outcomes add up to 1.
Using the context
Read carefully how many items or outcomes there are in total. A standard pack has 52 cards, with 4 of each value, so P(king) = 4/52 = 1/13. A fair die has 6 equally likely faces.
Worked examples
Example 1. A bag has 4 red, 7 blue and 9 green counters. One is chosen at random. Find P(blue).
- Total counters: 20.
- 7 are blue.
Answer: 7/20.
Example 2. Using the same bag, find P(not green).
- P(green) = 9/20.
- P(not green) = 1 − 9/20.
Answer: 11/20.
Example 3. A spinner lands on A, B, C or D. P(A) = 0.1 and P(B) = 0.36. P(D) is twice P(C). Find P(C) and P(D).
- P(C) + P(D) = 1 − 0.46 = 0.54.
- C + 2C = 3C = 0.54, so P(C) = 0.18.
Answer: P(C) = 0.18 and P(D) = 0.36.
Example 4. A fair die is rolled. Find the probability of rolling a factor of 6.
- Factors of 6: 1, 2, 3 and 6, which is 4 outcomes out of 6.
Answer: 4/6 = 2/3.
Example 5. A card is taken at random from a standard pack of 52. Find P(a king).
- There are 4 kings.
- 4/52.
Answer: 1/13.
Common mistakes
- Writing a probability as a ratio. Use a fraction, decimal or percentage.
- A probability greater than 1. Every probability is between 0 and 1.
- Using the wrong total. Count all the possible outcomes, not just two of the colours.
- Forgetting that outcomes add up to 1. Use 1 − (known probabilities) for the missing one.
- Not simplifying when asked. Give fractions in their simplest form if the question says so.
How basic probability is examined
Basic probability is Foundation content in both Edexcel specifications, 1MA1 and 4MA1, and it underpins every later probability topic. Questions ask you to:
- find a probability from equally likely outcomes
- find P(not A)
- find a missing probability using the fact that probabilities add up to 1
- mark a probability on a probability scale
The basic probability worksheet includes answers, and the GCSE probability revision engine gives unlimited practice.
Next, see mutually exclusive events and expected frequency.
Frequently asked questions
What is the probability of an impossible event?
0.
How do you find the probability of something not happening?
Subtract its probability from 1.
Can a probability be written as a ratio?
No. Use a fraction, decimal or percentage.