What you'll learn
Probability measures how likely an event is to occur, expressed as a number between 0 and 1. This topic appears in both Foundation and Higher tier OCR GCSE Mathematics papers, covering single events, combined events, tree diagrams, Venn diagrams, and conditional probability. You'll develop skills to calculate theoretical and experimental probabilities in contexts ranging from dice and cards to real-world scenarios.
Key terms and definitions
Probability — A numerical measure of how likely an event is to happen, ranging from 0 (impossible) to 1 (certain), often expressed as a fraction, decimal, or percentage.
Outcome — A possible result from a probability experiment or trial.
Sample space — The set of all possible outcomes for a particular experiment.
Event — A collection of one or more outcomes from a sample space.
Mutually exclusive events — Events that cannot happen at the same time; if one occurs, the other cannot.
Independent events — Events where the occurrence of one does not affect the probability of the other occurring.
Conditional probability — The probability of an event occurring given that another event has already occurred.
Expected frequency — The theoretical number of times an outcome should occur in a given number of trials, calculated as probability × number of trials.
Core concepts
Basic probability calculations
The probability of an event A is calculated using:
P(A) = Number of favourable outcomes / Total number of possible outcomes
The probability scale runs from 0 to 1:
- P(A) = 0 means the event is impossible
- P(A) = 1 means the event is certain
- P(A) = 0.5 means the event is equally likely to happen or not happen
For any event A, the probability that A does not occur is: P(A') = 1 - P(A)
where A' (read as "A prime" or "not A") represents the complement of event A.
Key rule: All probabilities in a sample space must sum to 1.
Listing outcomes systematically
For combined events, you must list all possible outcomes systematically. Use:
Sample space diagrams — A grid showing all possible combinations when two events occur together. For example, rolling two dice produces 36 possible outcomes arranged in a 6×6 grid.
Lists — Write ordered pairs or use systematic listing. When tossing two coins: HH, HT, TH, TT (4 outcomes).
Always count outcomes carefully. When order matters (like selecting a president then secretary from 5 people), you'll have 5 × 4 = 20 outcomes. When order doesn't matter (selecting any 2 people), you'll have fewer unique combinations.
Mutually exclusive and independent events
Mutually exclusive events cannot occur simultaneously. For example, rolling a 3 or a 5 on a single die — you cannot get both on one roll.
For mutually exclusive events A and B: P(A or B) = P(A) + P(B)
Independent events have probabilities that don't affect each other. Rolling a die twice — the second roll's outcome doesn't depend on the first.
For independent events A and B: P(A and B) = P(A) × P(B)
Common independent events in GCSE questions:
- Multiple coin tosses
- Replacing a card/ball before drawing again
- Rolling dice multiple times
- Spinning a spinner repeatedly
Tree diagrams
Tree diagrams show all possible outcomes for combined events across multiple stages. Each branch represents an outcome, with its probability written on the branch.
Construction rules:
- Draw branches for first event, label with probabilities (must sum to 1)
- From each outcome, draw branches for second event
- Continue for each subsequent event
- Probabilities on branches leaving the same point must sum to 1
Calculating with tree diagrams:
- To find P(specific path): multiply along the branches
- To find P(particular outcome from multiple paths): add the probabilities of all relevant paths
For example, with replacement (independent events), probabilities stay constant on each branch. Without replacement (dependent events), probabilities change based on previous outcomes.
Venn diagrams
Venn diagrams use overlapping circles to represent events and their relationships visually.
Key regions:
- Inside circle A only: items in A but not B
- Inside circle B only: items in B but not A
- Overlap (intersection): items in both A and B
- Outside both circles: items in neither A nor B
Notation:
- P(A ∩ B) = probability of A and B (intersection)
- P(A ∪ B) = probability of A or B or both (union)
- P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
The subtraction of P(A ∩ B) prevents double-counting the overlap.
Venn diagrams with frequency data: Write the number of items in each region. Start by filling the intersection, then work outwards to complete other regions using totals provided.
Conditional probability (Higher tier)
Conditional probability asks: "What's the probability of B given that A has already occurred?"
Notation: P(B|A) reads as "probability of B given A"
Formula: P(B|A) = P(A and B) / P(A)
This formula restricts the sample space to only those outcomes where A has occurred.
Example context: A bag contains 3 red and 2 blue balls. If you draw a red ball and don't replace it, the probability of drawing red again has changed — this is conditional probability.
From Venn diagrams: If you know events overlap, use the values from the diagram: P(B|A) = (number in A and B) / (total number in A)
Experimental probability and expected outcomes
Experimental probability (or relative frequency) uses actual results from trials:
Experimental probability = Number of times event occurred / Total number of trials
As the number of trials increases, experimental probability tends toward theoretical probability (the Law of Large Numbers).
Expected frequency predicts how many times an event should occur:
Expected frequency = Probability × Number of trials
For example, if P(heads) = 0.5 and you flip a coin 200 times, expect approximately 100 heads.
Questions often ask you to compare experimental and theoretical values, or to predict frequencies from probability.
Worked examples
Example 1: Mutually exclusive events with a spinner
Question: A spinner has 8 equal sections: 3 red, 3 blue, 1 yellow, 1 green.
(a) Find P(red) [1 mark] (b) Find P(red or blue) [2 marks] (c) The spinner is spun 200 times. How many times would you expect it to land on red? [2 marks]
Solution:
(a) P(red) = 3/8 ✓
(b) Red and blue are mutually exclusive P(red or blue) = P(red) + P(blue) ✓ = 3/8 + 3/8 = 6/8 = 3/4 ✓
(c) Expected frequency = probability × number of trials ✓ = 3/8 × 200 = 75 times ✓
Example 2: Tree diagram without replacement
Question: A box contains 5 milk chocolates and 3 dark chocolates. Sarah picks two chocolates at random without replacement.
(a) Complete the tree diagram [2 marks] (b) Calculate the probability both chocolates are milk chocolate [2 marks] (c) Calculate the probability Sarah picks one of each type [3 marks]
Solution:
(a) First pick: P(Milk) = 5/8, P(Dark) = 3/8 ✓ Second pick after milk: P(Milk) = 4/7, P(Dark) = 3/7 Second pick after dark: P(Milk) = 5/7, P(Dark) = 2/7 ✓
(b) P(both milk) = 5/8 × 4/7 ✓ = 20/56 = 5/14 ✓
(c) P(one of each) = P(Milk then Dark) + P(Dark then Milk) ✓ = (5/8 × 3/7) + (3/8 × 5/7) ✓ = 15/56 + 15/56 = 30/56 = 15/28 ✓
Example 3: Venn diagram and conditional probability (Higher)
Question: In a class of 30 students, 18 study French (F), 15 study Spanish (S), and 8 study both languages.
(a) Draw a Venn diagram to represent this information [2 marks] (b) Find P(F ∪ S) [2 marks] (c) Find the probability a student studies French given they study Spanish [2 marks]
Solution:
(a) Students studying both = 8 (in intersection) ✓ French only = 18 - 8 = 10 Spanish only = 15 - 8 = 7 Neither = 30 - 10 - 8 - 7 = 5 ✓
(b) P(F ∪ S) = (10 + 8 + 7)/30 ✓ = 25/30 = 5/6 ✓
(c) P(F|S) = P(F and S) / P(S) ✓ = (8/30) / (15/30) = 8/15 ✓
Alternatively: P(F|S) = (number in both) / (total in S) = 8/15
Common mistakes and how to avoid them
Adding instead of multiplying for independent events — Remember: "and" means multiply probabilities along branches; "or" means add probabilities of different paths. Check whether events happen together (multiply) or separately (add).
Forgetting to subtract P(A ∩ B) in the addition rule — When events aren't mutually exclusive, P(A ∪ B) = P(A) + P(B) - P(A ∩ B). The overlap gets counted twice if you just add, so subtract it once.
Not adjusting probabilities without replacement — When items aren't replaced, both the numerator and denominator change for subsequent events. If you pick a red ball from 3 red and 2 blue, the next pick is from 2 red and 2 blue (4 total), not the original 5.
Misreading Venn diagram regions — Start by filling in the intersection (both A and B), then calculate the "only A" and "only B" regions by subtracting. The value given for "study French" usually includes those who study both languages.
Writing probabilities greater than 1 — All probabilities must be between 0 and 1 inclusive. If your answer exceeds 1, you've made an error (often adding when you should multiply, or missing a step).
Not simplifying fractions — Examiners expect final answers as simplified fractions unless the question specifies decimals. 20/56 should be written as 5/14.
Exam technique for Probability
"Show all possible outcomes" requires a complete, systematic list or diagram. Use a sample space diagram for two dice, or write ordered pairs methodically. Marks are awarded for method, so show your working clearly even if the final list is obvious to you.
Check probabilities sum to 1 in the sample space or when finding complementary events. This provides a quick accuracy check. On tree diagrams, probabilities leaving each branch point must sum to 1.
Read "with/without replacement" carefully — This single word changes whether events are independent or dependent. Highlight or underline this word when reading questions to ensure you construct tree diagrams correctly.
Conditional probability notation — P(A|B) is different from P(A and B). The vertical bar means "given that," so you're working with a reduced sample space. Show the formula you're using to ensure method marks even if calculation errors occur.
Quick revision summary
Probability ranges from 0 to 1. Calculate using favourable outcomes divided by total outcomes. For mutually exclusive events, add probabilities (OR rule). For independent events, multiply probabilities (AND rule). Tree diagrams show combined events: multiply along branches, add across paths. Venn diagrams display overlapping events; remember to subtract the intersection when finding unions. Without replacement, probabilities change on subsequent selections. Expected frequency equals probability times number of trials. For conditional probability, restrict your sample space to given conditions and use P(B|A) = P(A and B) / P(A).