Probability is the maths of chance, and at 11+ level it is one of the friendlier topics on the paper — most questions come down to a single fraction or a short piece of yes/no logic. This guide gives you the complete framework: the probability scale, the core formula, worked examples, the trickier question types, and the exact ways children lose marks in a real exam. Work through it with your child and back it up with our free maths worksheets and timed mock papers.
Where probability appears in the 11+
Probability is more common in independent-school and Essex CSSE papers than in a standard GL Assessment paper, where it appears only occasionally. GL is now the dominant provider across most UK regions after CEM's separate 11+ tests were phased out around 2022-2023, so if your child is sitting a GL grammar-school test they may see one or two chance questions at most. If they are aiming at an independent senior school (often via the ISEB Common Pre-Test) or a CSSE Essex place, probability is far more likely — and CSSE's free-response format means the working matters, not just the letter. Either way, the skill is quick to master and worth banking. You can check what your target schools use in our grammar school directory.
In a GL paper the format is tight: typically 50 questions in 50 minutes, roughly a minute a question, multiple-choice with answers A to E marked on a separate OMR answer sheet, and no calculator. That means a probability question needs to be a 30-second job — spot the favourable outcomes, spot the total, simplify, and move on.
The probability scale
Every probability sits somewhere between 0 and 1. Nothing can be less than impossible or more than certain, so an answer above 1 is always wrong.
- 0 = impossible (for example, rolling a 7 on an ordinary six-sided dice)
- 1/4 = unlikely
- 1/2 or 0.5 = an even, 50-50 chance
- 3/4 = likely
- 1 = certain (for example, rolling a number less than 7 on a six-sided dice)
A useful check: the probability of an event happening plus the probability of it not happening always adds up to 1. If the chance of rain is 3/10, the chance of no rain is 7/10.
The core formula
Every basic 11+ probability question uses one rule:
Probability of an event = number of favourable outcomes ÷ total number of possible outcomes.
Write it as a fraction, then simplify. That is genuinely all most questions need.
Worked examples, step by step
Coin flip
What is the probability of flipping a head? Favourable outcomes: 1 (head). Total outcomes: 2 (head or tail). Probability = 1/2.
Dice roll
What is the probability of rolling an even number on a six-sided dice? Favourable: 3 (the 2, 4 and 6). Total: 6. Probability = 3/6, which simplifies to 1/2. Always give the simplified answer — examiners expect it, and in multiple-choice the simplified value is usually the option listed.
Bag of marbles
A bag holds 3 red, 4 blue and 5 green marbles. What is the probability of pulling out a red? Favourable: 3. Total: 3 + 4 + 5 = 12. Probability = 3/12 = 1/4. The single most common slip here is forgetting to add all three colours to get the total.
A reference table of common chances
Knowing these instantly saves precious seconds. Fluency in fraction-decimal-percentage equivalence (a core 11+ skill in its own right) makes probability much faster.
| Situation | Fraction | Decimal | Percentage |
|---|---|---|---|
| Coin lands heads | 1/2 | 0.5 | 50% |
| One specific number on a dice | 1/6 | 0.166... | ≈16.7% |
| An even number on a dice | 1/2 | 0.5 | 50% |
| A red card from a full pack | 1/2 | 0.5 | 50% |
| One specific card from 52 | 1/52 | 0.019... | ≈1.9% |
| Certain event | 1/1 | 1.0 | 100% |
Trickier question types
Without replacement
Watch for the words "does not put it back". A bag has 4 red and 6 blue marbles. A child pulls out one red and keeps it. What is the probability the next marble is also red? After removing one red, there are 3 reds left and 9 marbles in total, so the probability is 3/9 = 1/3. Both the favourable count and the total change — miss the changed total and the answer is wrong.
Two events happening together
For two independent events both happening, multiply the two probabilities. The chance of two heads in a row is 1/2 × 1/2 = 1/4. Rolling a six then another six is 1/6 × 1/6 = 1/36.
Either/or events
For the probability of one outcome or another (when they cannot both happen at once), add the probabilities. On a dice, the chance of a 1 or a 6 is 1/6 + 1/6 = 2/6 = 1/3.
"Not" questions
The probability of not rolling a 3 on a dice is 1 minus the probability of rolling a 3: 1 − 1/6 = 5/6. Reaching for the "1 minus" trick is often faster than counting five separate outcomes.
A method for every probability question
- Underline exactly what is being asked — heads, evens, red, or "not red".
- Count the favourable outcomes.
- Count the total number of possible outcomes.
- Write the fraction: favourable over total.
- Simplify fully, and check the answer is between 0 and 1.
For "and" questions multiply; for "or" questions add; for "not" questions subtract from 1; and for "without replacement" questions, update both numbers before you divide.
Difficulty progression by year group
A Year 4 or 5 child should be secure with the scale (impossible to certain), the basic formula, and single-fraction questions on coins, dice and bags. A Year 6 child sitting exam-level questions should also handle "without replacement", combined events, "not" questions, and probability expressed as a decimal or percentage. Build the timeline gradually using our revision hub so nothing is crammed the week before.
Exam traps that cost marks
- Wrong total. Forgetting to add every group (all colours of marble, all sides of a spinner) before dividing.
- Not simplifying. Writing 3/6 when the answer sheet lists 1/2 — the un-simplified fraction may not be an option.
- Ignoring the last sentence. A worded problem often switches at the end from "red" to "not red", or asks for a percentage rather than a fraction.
- Missing "without replacement". Forgetting that removing an item changes both the top and bottom of the fraction.
- Adding when you should multiply. "Both" means multiply; "either" means add.
Frequently asked questions
Is a calculator allowed for 11+ probability questions?
No. GL and most 11+ papers are non-calculator, so all probability working is done by hand. The numbers are deliberately kept simple, so quick fraction skills matter more than heavy arithmetic.
How should the answer be written?
Nearly always as a simplified fraction, though some papers accept a decimal or percentage. Simplify fully, and never leave an answer greater than 1 — that is always a mistake.
How much probability will actually come up?
Only a little in a standard GL grammar paper, but more in CSSE Essex and independent-school papers. It is a fast topic to secure, so it is well worth a short revision session for the marks it can guarantee.
Ready to practise? Our probability drills are graded from foundation to exam-level, all completely free.