KS3 Mathematics · National Curriculum · Probability
Mini-Lesson
Probability
This mini-lesson walks you through the whole KS3 Probability strand: the 0–1 probability scale, working out the chance of a single event, experimental vs theoretical probability, sample spaces & Venn diagrams, and why the probabilities of all outcomes add up to 1.
Every probability lives somewhere on this line from 0 (impossible) to 1 (certain).
Work through each screen, answer the questions as you go (some are worded, some are calculations) and collect ⭐ stars. Press Start when you're ready.
The probability scale
How likely? From 0 to 1
Probability measures how likely an outcome is. We can describe it in words or as a number between 0 and 1 (as a fraction, decimal or percentage):
Words like "unlikely" and "even chance" line up with numbers on the same scale.
Watch out: a probability can never be less than 0 or more than 1. An answer like "1.5" or "−0.2" or "120%" must be wrong.
Quick check
Reading the scale
?The probability that it snows in London tomorrow in July is 0.02. Which word describes this best?
Single-event probability
Counting equally likely outcomes
When every outcome is equally likely, the probability of an event is:
P(event) = favourable ÷ totalP(event) = number of favourable outcomes ÷ total number of possible outcomes
For a fair six-sided dice there are 6 equally likely outcomes. Rolling a 4 is just 1 of them, so P(4) = 1/6.
Worked example
A fair dice is rolled. Find P(an even number).
Even numbers are 2, 4, 6 → that's 3 favourable outcomes out of 6.
P(even) = 3/6 = 1/2 (0.5)
Calculate
Your turn — one spinner
1A bag holds 3 red, 4 blue and 1 green counter (8 counters in total). One is taken at random. What is P(blue)? Give your answer as a decimal.
Hint: P(blue) = 4 ÷ 8.
Fair or not?
"Two things can happen" ≠ 50–50
A common mistake is to think that if there are two possible results, each must have probability ½. That is only true when the outcomes are equally likely.
You either win the lottery or you don't — but that is not a 50–50 chance! Winning is extremely unlikely.
Different-sized sections mean the outcomes are not equally likely.
Watch out: only split the total evenly when the outcomes really are the same size / equally likely.
Quick check
Is it 50–50?
?A spinner has 3 equal sections coloured red, red and blue. Priya says "There are two colours, so P(blue) = ½." Why is she wrong?
Outcomes sum to 1
They must all add up to 1
Outcomes that cannot happen at the same time are called mutually exclusive. For a single trial, the probabilities of all the possible outcomes always add up to 1.
P(not A) = 1 − P(A)because something either happens or it doesn't, and the two add to 1
Worked example
The probability a bus is late is 0.15. What is the probability it is not late?
P(not late) = 1 − 0.15 = 0.85
Watch out: this only works when the outcomes are mutually exclusive (they can't both happen). Rolling a "3" and rolling an "even" on one dice are not mutually exclusive with all outcomes — be careful what you're subtracting.
Calculate
Your turn — the missing probability
2A biased coin lands on Heads with probability 0.7. The only other outcome is Tails. What is P(Tails)? Give a decimal.
Hint: the two probabilities must add to 1, so P(Tails) = 1 − 0.7.
Calculate
Your turn — three colours
3A spinner can land on red, blue or green only. P(red) = 0.4 and P(blue) = 0.35. What is P(green)? Give a decimal.
Hint: all three must total 1, so P(green) = 1 − 0.4 − 0.35.
Experimental probability
Learning from real trials
Sometimes we can't just count equally likely outcomes — a spinner might be biased. So we do the experiment and record what actually happens. The relative frequency is:
relative frequency = successes ÷ trialsrelative frequency = number of times it happened ÷ total number of trials
Worked example
A drawing pin is dropped 200 times and lands "point up" 130 times.
Experimental P(point up) = 130 ÷ 200 = 0.65
The more trials you do, the closer the relative frequency usually gets to the true probability.
Experimental vs theoretical
Two ways to a probability
Theoretical probability — worked out by counting equally likely outcomes (e.g. P(6) = 1/6 for a fair dice).
Experimental probability — measured from actual results (relative frequency).
For a fair object, experimental results wobble around the theoretical value and settle closer to it as trials increase:
With more trials the relative frequency (blue) settles towards the theoretical value (green).
Watch out: a fair coin can still give 7 heads in 10 throws. Short experiments are noisy — that doesn't prove it's biased.
Calculate
Your turn — expected number
4The experimental probability a bus is late is 0.2. Out of 150 buses, how many would you expect to be late?
Hint: expected number = probability × number of trials = 0.2 × 150.
Play
Mini-game 1 — Place it on the scale
Tap the word that best matches each probability.
Sample spaces
Listing every outcome
A sample space is a list (or grid) of all possible outcomes. For two events, a grid keeps you organised so you don't miss any.
Here is the sample space for the total when two fair dice are rolled:
Two dice give 6 × 6 = 36 equally likely outcomes. A total of 7 appears 6 times, so P(7) = 6/36 = 1/6.Calculate
Your turn — reading the grid
5Using the two-dice grid, how many of the 36 outcomes give a total of 10? (Count them in the grid.)
Hint: look for every 10 in the grid — 4+6, 5+5, 6+4.
Quick check
Two-dice probability
?From the grid of 36 outcomes, what is the probability the two dice total 12?
Sets & Venn diagrams
Overlaps: Venn diagrams
A Venn diagram sorts things into sets. The overlap (the intersection) holds things in both sets; the whole of both circles is the union.
7 students study both (the intersection). 12 + 7 + 6 = 25 study at least one (the union); 5 study neither.
Check: 12 + 7 + 6 + 5 = 30, the whole class. Everyone must appear exactly once.
Calculate
Your turn — from the Venn diagram
6Using the French/Spanish Venn diagram, one of the 30 students is chosen at random. How many study only French (French but not Spanish)?
Hint: "only French" is the part of the French circle that is NOT in the overlap.
Quick check
Union from the Venn
?A student is picked at random from the 30. What is the probability they study at least one language (French or Spanish or both)?
Play
Mini-game 2 — Mutually exclusive?
For one roll of a single dice, tap a pair, then drop it in the right box. Mutually exclusive = the two can't both happen at once.
🚫 Mutually exclusive
🔗 Can both happen
Calculate
Your turn — adding exclusive probabilities
7A fair dice is rolled. P(rolling a 1) = 1/6 and P(rolling a 2) = 1/6. Because you can't roll both at once, P(1 or 2) = 1/6 + 1/6. Write this as a fraction with denominator 6 (type just the top number).
/6
Hint: for mutually exclusive outcomes you can ADD the probabilities: 1/6 + 1/6 = 2/6.
Recap
The ideas to know
Scale: probability runs 0 (impossible) → 1 (certain).
Single event: P = favourable ÷ total (equally likely only).
Sum to 1: all outcomes add to 1; P(not A) = 1 − P(A).
Experimental: relative frequency = successes ÷ trials.
Sample space: list/grid of all outcomes (two dice = 36).
Venn / sets: intersection = both; union = at least one.
You've covered the whole KS3 Probability strand — the scale, single & combined events, experimental vs theoretical, sample spaces, Venn diagrams and mutually exclusive outcomes. Press Finish to see your score.
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