Basic Probability Notes
Edexcel GCSE Maths: Revision notes
Key facts
- Probability runs from 0 (impossible) to 1 (certain); it is never negative or above 1.
- For equally likely outcomes, .
- Expected number probability number of trials.
- Relative frequency from an experiment estimates probability; more trials give a better estimate.
- Probabilities of all outcomes add to 1, so .
The probability scale
Probability measures how likely an event is, from 0 (impossible) to 1 (certain).
It can be written as a fraction, a decimal or a percentage.
- 0: impossible
- Close to 0: unlikely
- 0.5: even chance
- Close to 1: likely
- 1: certain
A probability can never be negative or greater than 1.
Which of these cannot be a probability?
Equally likely outcomes
When all outcomes are equally likely, divide the favourable outcomes by all possible outcomes.
This assumes fair, random outcomes, such as a fair coin or an unbiased die.
From a probability you can find the expected number of successes in trials. It is an average, not a promise.
- Probability
- Expected numberprobability × number of trials
Worked example
A fair die is rolled 60 times. How many 6s would you expect?
- 1
P(6) = 1/6.
- 2
Expected = 1/6 × 60.
A fair spinner has 5 equal sections, 2 red. It is spun 100 times. How many reds do you expect?
Relative frequency
Relative frequency is worked out from results, and it settles towards the true probability as trials increase.
Theoretical probability comes from equally likely outcomes, e.g. for a die.
Relative frequency comes from an experiment: number of times the event occurs ÷ number of trials.
It is only an estimate. The more trials, the better the estimate.
| Theoretical | Relative frequency | |
|---|---|---|
| Based on | Equally likely outcomes | Results of an experiment |
| Key point | Calculated before you start | More trials give a better estimate |
Theoretical
- Based on:
- Equally likely outcomes
- Key point:
- Calculated before you start
Relative frequency
- Based on:
- Results of an experiment
- Key point:
- More trials give a better estimate
Worked example
A coin is flipped 100 times and lands heads 47 times. Find the relative frequency of heads.
- 1
47 ÷ 100 = 0.47.
- 2
Theoretical probability is 0.5, so this is close.
Which experiment gives the best estimate of the probability a drawing pin lands point up?
Probabilities sum to 1
The probabilities of all possible outcomes add up to 1, so the chance of something not happening is 1 minus its probability.
For a set of outcomes that must include one that happens (exhaustive), the probabilities add to exactly 1.
Events that cannot happen at the same time are mutually exclusive.
- P(not A)1 − P(A)
Worked example
The probability that it rains tomorrow is 0.3. Find the probability that it does not rain.
- 1
1 − 0.3 = 0.7.
The probability a bus is late is 0.15. What is the probability it is not late?
Two-way tables
A two-way table sorts data by two categories at once, so probabilities can be read from it.
Find the row or column total for the event, then divide the relevant frequency by the correct total (usually the grand total).
Two-way tables list every combination without missing any.
- Boys
- Girls
| Boys | Girls | All | |
|---|---|---|---|
| French | 8 | 6 | 14 |
| Spanish | 7 | 9 | 16 |
| Total | 15 | 15 | 30 |
Boys
- French:
- 8
- Spanish:
- 7
- Total:
- 15
Girls
- French:
- 6
- Spanish:
- 9
- Total:
- 15
All
- French:
- 14
- Spanish:
- 16
- Total:
- 30
Worked example
A student is chosen at random from the 30. Find the probability they are a girl who studies Spanish.
- 1
Girls studying Spanish: 9.
- 2
9 out of 30.
From the same table, what is the probability a student studies French?
Try an exam question
A fair 5-sided spinner is numbered 1 to 5. (a) Find the probability of spinning a prime number. (b) The spinner is spun 200 times. How many times would you expect a prime number?
[4 marks]
- [1]Prime numbers are 2, 3 and 5: 3 outcomes out of 5.
- [1]P(prime) = 3/5.
- [1]Expected = 3/5 × 200.
- [1]120 times.
That's the notes covered.
Carry on to the next subtopic.