Probability: likelihood, outcomes and combined events
What Probability tests
Probability tests how likely events are, using sample spaces, relative frequency, expected outcomes, the complement rule, combined events and tree diagrams. Foundation questions use coins, dice, spinners, bags of counters and two-way tables. Answers should be fractions, decimals or percentages between 0 and 1, with method shown.
Core rules
- P(event) = number of favourable outcomes ÷ total number of equally likely outcomes.
- The complement rule: P(not A) = 1 - P(A).
- For mutually exclusive events, add probabilities: P(A or B) = P(A) + P(B).
- For independent events, multiply probabilities: P(A and B) = P(A) × P(B).
- Expected frequency = probability × number of trials.
- Relative frequency = number of times the event happened ÷ total number of trials.
Worked example
A bag has 3 red and 5 blue counters. Two counters are taken without replacement. Find the probability that both are red. First draw: P(R) = 3/8. After removing one red, 2 red and 5 blue remain, so P(R second | R first) = 2/7. Multiply: 3/8 × 2/7 = 6/56 = 3/28. With replacement, the second probability would stay 3/8, giving 9/64. The words without replacement change the denominator.
Traps that cost marks
- Adding probabilities when events happen together instead of multiplying.
- Forgetting to reduce the denominator after a counter is removed without replacement.
- Giving a probability greater than 1 or negative.
- Confusing theoretical probability with relative frequency from an experiment.