GCSE Foundation — Probability
Every probability statement a GCSE Foundation student may be asked is here; together they are 7.5% of the assessment. 8 statements make up probability here. Each has its own page with the specification wording, the common mistakes, and questions. The ground it covers runs through frequency of outcomes, tables and frequency trees, expected outcomes and fairness, relative frequency and the probability scale, probabilities of exhaustive events sum to one, empirical samples and sample size and sets, Venn diagrams and tree diagrams. Everything on this page is Foundation content, which means a Higher student needs it too — Higher is Foundation plus more, not instead of. With 50% AO1 and 50% across AO2 and AO3, fluency alone is not enough — the paper repeatedly asks what your answer means. The bulk of it sits comfortably on the non-calculator paper; a calculator quietly hides the gap until the exam.
- P1 — Frequency of outcomes, tables and frequency trees
- P2 — Expected outcomes and fairness
- P3 — Relative frequency and the probability scale
- P4 — Probabilities of exhaustive events sum to one
- P5 — Empirical samples and sample size
- P6 — Sets, Venn diagrams and tree diagrams
- P7 — Possibility spaces for single and combined events
- P8 — Independent and dependent combined events
Free · no sign-up · 224 questions in this area
Probability at GCSE Foundation: the 8 DfE statements
Every question on this site is filed against one of these statements. Pick one to practise it on its own.
Sample probability questions for GCSE Foundation
- The probability that Amelia's bus is late on a school morning is 9/10. Write this probability as a percentage.(a)90%(b)9%(c)10%(d)0.9%
Show the answer
90% — Method: turn the fraction into a decimal by dividing the numerator by the denominator, then multiply the decimal by 100 to turn it into a percentage. Working: 9 ÷ 10 = 0.9, and 0.9 × 100 = 90. Answer: 90%, which is the same probability written in a third form, and all three forms sit at the same point on the 0 to 1 scale. The distractors: 9% comes from reading the numerator of the fraction straight off as the percentage; 10% comes from reading the denominator straight off as the percentage; 0.9% comes from correctly finding the decimal 0.9 and then writing a percentage sign after it without multiplying by 100. - Four bags each contain counters, some of which are winning counters. The probability of taking a winning counter is 5/12 from the red bag, 3/8 from the blue bag, 7/24 from the green bag and 1/3 from the yellow bag. Work out which bag has the smallest probability of giving a winning counter.(a)The blue bag (3/8)(b)The red bag (5/12)(c)The green bag (7/24)(d)The yellow bag (1/3)
Show the answer
The green bag (7/24) — Method: fractions can only be ordered once they share a denominator, so rewrite all four over the lowest common denominator and compare the numerators. Working: the lowest common denominator of 12, 8, 24 and 3 is 24, and scaling gives 5/12 = 10/24, 3/8 = 9/24, 7/24 stays as it is, and 1/3 = 8/24; the numerators are then 10, 9, 7 and 8. Answer: the smallest numerator is 7, so the green bag, with 7/24, is the least likely and sits furthest to the left on the 0 to 1 scale. The distractors: the red bag (5/12) comes from finding the largest of the four probabilities instead of the smallest; the blue bag (3/8) comes from scaling 3/8 by changing only the denominator to 24, which turns it into 3/24 and makes it look the smallest; the yellow bag (1/3) comes from comparing numerators alone and assuming the fraction with the numerator 1 must be the smallest. - A card is taken at random from an ordinary pack of 52 playing cards. The probability that the card is a diamond is 1/4. Work out the probability that the card is not a diamond.(a)1/2(b)3/4(c)1/4(d)3/52
Show the answer
3/4 — Method: a card either is a diamond or is not a diamond, so those two outcomes form an exhaustive set and their probabilities add to 1; subtract the given probability from 1. Working: P(diamond) = 1/4, so P(not a diamond) = 1 − 1/4; writing 1 as 4/4 gives 4/4 − 1/4. Answer: 3/4. The distractors: 1/4 comes from giving back the probability that the card is a diamond instead of its complement; 1/2 comes from reading 'not a diamond' as 'not a red card' and halving the pack; 3/52 comes from doing the subtraction 4 − 1 = 3 on the suits but then writing that 3 over the 52 cards in the pack instead of over the 4 suits. - The probability that a pupil chosen at random has a nut allergy is 1/10. There are 30 pupils in a class. Work out how many of the 30 pupils would be expected to have a nut allergy.(a)1 pupil(b)3 pupils(c)27 pupils(d)10 pupils
Show the answer
3 pupils — Method: an expected frequency is the probability multiplied by the number of trials, so multiply the probability by the number of pupils. Working: 30 × 1/10 means finding one tenth of 30, and 30 ÷ 10 = 3. Answer: 3 pupils would be expected to have a nut allergy. The distractors: 27 pupils comes from working out how many are expected NOT to have the allergy, 30 − 3, instead of how many are; 10 pupils comes from reading the 10 in the fraction 1/10 as the number of pupils; 1 pupil comes from reading the numerator of the fraction as the expected number. - The probability that a spinner lands on blue is 2/5. Write this probability as a percentage.(a)20%(b)40%(c)2%(d)4%
Show the answer
40% — Method: a fraction becomes a percentage by scaling it so that its denominator is 100, or equivalently by dividing the numerator by the denominator and multiplying the decimal by 100. Working: 5 × 20 = 100, so the numerator is scaled by 20 as well, 2 × 20 = 40, giving 40/100; the same figure comes from 2 ÷ 5 = 0.4 and 0.4 × 100 = 40. Answer: 40%. The distractors: 20% comes from dividing 100 by the denominator alone, 100 ÷ 5 = 20, which is the percentage for 1/5 and never uses the numerator 2; 4% comes from finding the decimal 0.4 correctly and then writing the 4 down as the percentage instead of multiplying by 100; 2% comes from scaling only the denominator up to 100 and leaving the numerator as it was, giving 2/100.
Frequently asked questions
How much of the GCSE Foundation paper is probability?
7.5% of the total marks, and the three papers each carry an equal share of the qualification, so it is spread across all of them rather than concentrated in one.
How many topics are there in probability at GCSE Foundation?
8 DfE content statements, coded P1 to P8 within this area. Every question in the bank is tagged to one of them.
Is any probability content on this page Higher only?
No. Foundation pages only show statements a Foundation entry may be asked. The Higher-only probability material is on the GCSE Higher page for this area.
Can I use a calculator for probability questions?
Paper 1 is non-calculator; Papers 2 and 3 allow one. Of the 8 statements in this area, 4 are naturally non-calculator and 0 naturally calculator, with the rest workable either way — so practise both.
What kind of marks does probability carry?
The GCSE Foundation split is 50% AO1 (use and apply standard techniques), 25% AO2 (reason, interpret and communicate) and 25% AO3 (solve problems in and out of context). Questions in this area are set across all three.
Are these past paper questions?
No. Every question at /gcse-foundation is original, written to the DfE content statements and checked before it is published. We do not host past papers or mark schemes.
Do I need an account?
No — you can start practising straight away. Progress is saved automatically in your browser.
Is it free?
Yes. The questions, worked answers and printable worksheets are all free, with no adverts.
Tips for probability at GCSE Foundation
- The probability of an event is the number of favourable outcomes divided by the total number of equally likely outcomes. It is always between 0 (impossible) and 1 (certain).
- Probabilities of all the outcomes add to 1, so P(not A) = 1 − P(A). It is often easier to work out the probability that something does not happen and subtract from 1.
- For two independent events, such as two coin flips, the probability that both happen is the product: P(A and B) = P(A) × P(B).
- Use a sample space or a table for two-stage experiments. Rolling two dice gives 6 × 6 = 36 outcomes, not 6.
- Tell the difference between picking with replacement (the probabilities stay the same) and without replacement (the numerator and denominator both change at the second pick).
Common mistakes (and how to avoid them)
- Adding probabilities for 'and' when you should multiply, and writing P(A) + P(B) for the probability that both events happen.
- Keeping the same denominator without replacement — writing 5/20 twice instead of 5/20 and then 4/19.
- Getting a probability greater than 1 or below 0 and not stopping to check. Every valid probability lies between 0 and 1.
- Treating outcomes as equally likely when they are not, for example the totals when two dice are rolled, where 7 is far more common than 2.
Worked examples
- The flips are independent, and P(heads) = ½ each time.
- P(heads and heads) = ½ × ½.
- ½ × ½ = ¼.
- First pick: 3 red out of 5, so P = 3/5.
- Without replacement there are now 2 red out of 4, so the second pick has P = 2/4 = ½.
- P(both red) = 3/5 × ½ = 3/10.
- There are 6 × 6 = 36 equally likely outcomes.
- A total of 12 comes from only one outcome: 6 and 6.
- Probability = 1/36.
Probability is the mathematics of uncertainty, and it is everywhere from weather forecasts to insurance. At Foundation you work with two-stage experiments, independent events, and picking with and without replacement. The key is to identify the structure before you calculate: are the stages independent? Is there replacement? A tree diagram or a sample-space table is your best friend because it lays out every outcome clearly. And always check that the answer lands between 0 and 1.
More GCSE Foundation maths:
← Every GCSE Foundation content area