GCSE Maths · Probability

Probability

Single events, two events, and tree diagrams.

200 GCSE-style practice questions

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Probability, explained point by point

Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.

  1. Use the probability scale from 0 to 1, and the fact that all outcomes sum to 1

    Probabilities live between 0 (impossible) and 1 (certain), as fractions, decimals or percentages. The probabilities of all possible outcomes add to 1, so P(not raining) = 1 − P(raining). "Not" questions are usually one subtraction.

  2. List outcomes systematically and use sample space diagrams

    Work through outcomes in a fixed order so none are missed: for two coins - HH, HT, TH, TT. A sample space diagram grids out two events, like the 36 results of rolling two dice, and probabilities are read off by counting cells.

  3. Calculate probabilities of equally likely outcomes

    When outcomes are equally likely, probability = favourable ÷ total: the chance of a prime on one dice roll is 36 = 12 (from 2, 3, 5). The formula fails for UNequal outcomes - a biased spinner needs the given probabilities instead.

  4. Estimate probability from relative frequency, and calculate expected frequency

    Relative frequency = times it happened ÷ trials: 13 heads in 40 flips estimates P(heads) ≈ 0.325, and more trials make the estimate more reliable. Expected frequency runs the other way: P × number of trials, so 0.3 × 200 = 60 expected wins.

  5. Use two-way tables, frequency trees and Venn diagrams

    These organise counts so probabilities can be read off. Fill in what is given, use row and column totals to deduce the gaps, then divide by the right total. In Venn diagrams, place the overlap FIRST and subtract outwards.

  6. Draw and use tree diagrams for combined events

    Each set of branches shows one event, with probabilities on the branches summing to 1. MULTIPLY along a path for "and"; ADD finished paths for "or". P(one head in two flips) = 12×12 + 12×12 = 12 - two paths give exactly one head.

  7. Calculate conditional probabilities, including without replacementHigher only

    Once the first item is taken and not replaced, the pool shrinks: after drawing a red from 5 red and 3 blue, the next draw is from 4 red and 3 blue - second-branch probabilities change. "Given that" questions divide by the probability of the known condition instead of the whole.

Practice

Try a Probability question

A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.

Two fair six-sided dice are rolled and their scores are added. What is the probability that the total is 9?

  1. 112
  2. 136
  3. 19
  4. 111
Show the answer and the working

Answer: 19

There are 36 equally likely pairs, and four of them total 9: (3,6), (4,5), (5,4) and (6,3).

  1. Two dice give 6 × 6 = 36 equally likely pairs.
  2. List the pairs totalling 9: (3,6), (4,5), (5,4), (6,3).
  3. That is 4 favourable pairs.
  4. P(total 9) = 436 = 19.

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