Basic Probability Notes

AQA GCSE Maths: Revision notes

Key facts

Probabilities lie between 0 and 1 and can be fractions, decimals or percentages.
Expected outcomes = probability $\times$ number of trials.
Relative frequency = successful trials $\div$ total trials; it nears the theoretical probability as trials increase.
The probabilities of an exhaustive set of outcomes sum to 1.
A possibility space lists every equally likely outcome so you can count favourable ones.

Recording experiments

Record outcomes in tables and frequency trees, then use probability to predict how many times an event should occur.

The results (outcomes) of an experiment can be recorded in tables and frequency trees.

Using fairness and equally likely events, you can calculate the expected outcomes of future trials, such as the number of heads in 100 tosses of a fair coin.

Expected outcomes in 100 tosses of a fair coin: probability × number of trials.

Expected outcomesprobability $\times$ number of trials

Worked example

A fair coin is tossed 50 times. How many heads should you expect?

1
The probability of heads is $0.5$ .
2
Expected heads $= 50 \times 0.5$ .

A fair six-sided die is rolled 120 times. How many sixes would you expect?

Relative frequency

Relative frequency comes from real results and gets closer to the theoretical probability with more trials.

Relative frequency is successful trials divided by total trials. More trials means random variation averages out, so the value settles near the theoretical probability.

Probabilities are on a scale from 0 (impossible) to 1 (certain).

Relative frequency $\dfrac{\text{successful trials}}{\text{total trials}}$

Worked example

A coin is tossed 40 times and lands on heads 18 times. Find the relative frequency of heads.

1
Successful trials: 18. Total trials: 40.
2
Relative frequency $= \dfrac{18}{40} = 0.45$ .

Why do experiments with more trials give better estimates of probability?

Sets of outcomes

The probabilities of all possible, mutually exclusive outcomes add up to 1, so you can find a missing probability.

An exhaustive set of outcomes covers every possibility, so the probabilities sum to 1. Mutually exclusive events cannot happen at the same time.

If events are exhaustive and mutually exclusive their probabilities also sum to 1. Use this to find a missing probability.

Illustrative example with P(A) = 0.3: P(A) + P(not A) = 1.

All outcomesprobabilities sum to 1
Not happening $P(\text{not } A) = 1 - P(A)$

Worked example

If $P(\text{rain}) = 0.3$ , find $P(\text{no rain})$ .

1
Rain and no rain are exhaustive and mutually exclusive, so the probabilities sum to 1.
2
$P(\text{no rain}) = 1 - 0.3$ .

A bag has red, blue and green counters. P(red) = 0.2 and P(blue) = 0.5. What is P(green)?

Possibility spaces

Tables, grids and Venn diagrams list outcomes; a possibility space lets you count favourable outcomes out of the total.

A possibility space lists every equally likely outcome of one or combined experiments. Then $P(\text{event}) = \dfrac{\text{favourable outcomes}}{\text{total outcomes}}$ .

Two dice have 36 equally likely outcomes. Draw a 6 by 6 table of totals and count: 6 of the 36 give a total of 7, so $P = \frac{1}{6}$ .

Ways to make each total with two dice (36 outcomes in all).

Using the table of totals for two dice, what is the probability of a total of 7?

Try an exam question

A bag contains only red, blue and green counters. The probability of red is 0.2 and the probability of blue is 0.5. Work out the probability of green. A counter is taken from the bag and replaced 80 times. How many times should you expect blue?

[4 marks]

That's the notes covered.

Carry on to the next subtopic.