Probability ToolkitEdexcel IGCSE Maths: Revision notes

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, decimal or percentage.

All probabilities for a single event satisfy: $0 \leq P(\text{event}) \leq 1$

A probability scale (a simple 0-to-1 number line) is used to show roughly how likely an event is: near 0 = unlikely, $\frac{1}{2}$ = evens, near 1 = likely.

For an experiment where all outcomes are equally likely, theoretical probability is calculated without doing any trials: $P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}$

Example: rolling a fair six-sided die, $P(\text{rolling a 5}) = \frac{1}{6}$.

When outcomes are not equally likely, or we have no theoretical model, we estimate probability from experiments using relative frequency: $\text{relative frequency} = \frac{\text{number of times the event occurred}}{\text{total number of trials}}$

Relative frequency is only an estimate of the true probability. The more trials carried out, the more reliable the estimate becomes — this is called the law of large numbers in everyday exam language ("the larger the number of trials, the better the estimate of probability").

A relative frequency table lets you track how an estimate changes as more trials are added, and questions often ask you to use relative frequency to predict an expected number of future outcomes: $\text{expected number} = \text{relative frequency} \times \text{number of future trials}$

Two events are mutually exclusive if they cannot both happen at the same time (e.g. rolling a 2 and rolling a 5 on the same die roll).

For mutually exclusive events, probabilities add: $P(A \text{ or } B) = P(A) + P(B)$

A key exam idea: the probabilities of all mutually exclusive outcomes in a sample space sum to exactly 1: $P(A) + P(B) + P(C) + \ldots = 1$

This is often used to find a "missing" probability: if $P(\text{red}) = 0.3$ and $P(\text{blue}) = 0.45$, and red, blue and green are the only mutually exclusive outcomes, then $P(\text{green}) = 1 - 0.3 - 0.45 = 0.25$.

The probability of an event not happening is: $P(\text{not } A) = 1 - P(A)$

Systematic listing means listing all possible outcomes of one or more events in a clear, ordered way so that none are missed and none are repeated.

Useful systematic methods:

• Two-way tables for combining two events (e.g. two dice, or coin + spinner)
• Sample space diagrams (grids) showing every combined outcome
• Ordered lists, fixing one item and cycling through the rest (e.g. listing all two-digit numbers from digits 1, 2, 3 without repeats: 12, 13, 21, 23, 31, 32)
• The product rule for counting: if one event has $m$ outcomes and a second independent event has $n$ outcomes, the total number of combined outcomes is $m \times n$.

Systematic listing is essential before calculating theoretical probability for combined events, since you need the correct total number of outcomes in the denominator.