Probability & Statistics
Basic probability, complement, expected value, combined events, Venn diagrams.
This page is a sequence, not a document: try a question cold, learn the methods, drill until they’re automatic, then prove it under a clock. Skipping straight to reading is the least effective thing you can do here — that’s not opinion, it’s the testing effect.
Try one first
Before any teaching — a real question tells you honestly where you stand.
| Mode | Count |
|---|---|
| Bus | 55 |
| Train | 60 |
| Car | 40 |
One commuter is chosen at random. What is the probability they take the Train?
The methods
Every probability & statistics pattern the tests use — the method in a few lines, then a worked example.
Probability from a table
Probability = favourable count ÷ total count (not ÷ the rest — that's odds).
Worked example
A survey of 250 commuters records their main mode of travel: Bus 90, Train 110, Car 50. One commuter is chosen at random. What is the probability they travel by train?
Answer: 44.0%
Probability = favourable ÷ total = 110/250 = 44.0%. The 78.6% foil divides by the non-train commuters (odds, not probability); 36% and 20% use the wrong mode; 45.8% uses a wrong total.
Venn diagrams
At least one = A + B − both (subtract the overlap once); neither = total − that.
Worked example
In a group of 80 people, 45 like tea, 38 like coffee, and 12 like both. How many like neither?
Answer: 9
Those who like at least one = 45 + 38 − 12 = 71 (subtract the overlap once). Neither = 80 − 71 = 9. The 71 foil gives 'at least one' instead of 'neither'; 35 and 42 remove only one set.
Expected value
Expected value = Σ(outcome × its probability).
Worked example
A game pays £10 with probability 0.2, £5 with probability 0.3, and £0 with probability 0.5. What is the expected winning?
Answer: £3.50
Expected value = Σ(prize × probability) = 10×0.2 + 5×0.3 + 0×0.5 = 2 + 1.5 = £3.50. The £5 foil averages the prizes (unweighted); £15 just sums them.
Combined probability
For independent events, P(A and B) = P(A) × P(B).
Complement rule
Complementary events sum to 1: P(not A) = 1 − P(A).
Worked example
The probability that it rains tomorrow is 0.35. What is the probability that it does NOT rain?
Answer: 0.65
Complementary events sum to 1, so P(no rain) = 1 − 0.35 = 0.65. The 0.35 foil repeats the given probability; 1.35 adds instead of subtracting.
Drill it
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Prove it under a clock
A skill isn't test-ready until it survives time pressure, mixed with everything else.
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