Probability measures how likely something is, on a scale from 0 (impossible) to 1 (certain). The basic rule is simple — count the outcomes you want, divide by the total. From there you build up the language of events: the complement (not happening), the intersection (both happening, “and”), and the union (either happening, “or”). Get these symbols and the two combining formulas down, and most of the unit falls into place.
📘 What you need to know
Sample space U: all possible outcomes. n(U) = total number of outcomes; n(A) = outcomes in event A.
UnionA ∪ B = “AorB or both” (at least one occurs).
Union formula: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Basic probability
For equally likely outcomes, probability is just a count. List or table the sample space, count what you want, divide by the total.
Theoretical probabilityP(A) = n(A)n(U)in the formula booklet ✓
Expected number of occurrences = np. If you roll a fair dice 60 times, you expect 60 × 16 = 10 sixes — on average, not exactly.
The complement: A′
The complement A′ is everything exceptA. Since one of A or A′ must happen, their probabilities sum to 1 — often the quickest route to an answer.
ComplementP(A) + P(A′) = 1 ⇒ P(A′) = 1 − P(A)
in the formula booklet ✓
🧠 Memory aid — “1 minus”
Whenever a question says “at least one” or “not”, reach for the complement: P(at least one) = 1 − P(none). It’s almost always faster than adding up every winning case.
Intersection (AND) and union (OR)
Two events can combine in two ways. ∩ means both (“and”); ∪ means at least one (“or”). The overlap is the key.
Intersection ∩
A and B
Both events happen together — the overlap region of the two circles.
Union ∪
A or B (or both)
At least one happens — everything inside either circle.
intersection vs union
Intersection = just the overlap. Union = both circles combined.
Union (addition) formulaP(A ∪ B) = P(A) + P(B) − P(A ∩ B)
in the formula booklet ✓
🤔 Why subtract P(A ∩ B)?
When you add P(A) and P(B), the overlap (where both happen) gets counted twice — once in each. Subtracting P(A ∩ B) removes the double-count, so the union is correct. If the events can’t both happen (no overlap), that term is 0 and you just add.
Worked examples
WE 1
List a sample space with a table
Spinner A has sides 1, 4, 9; spinner B has sides 2, 3, 5, 7. A two-digit number is formed (A = first digit, B = second). List all possible numbers.
use a two-way table (A down, B across)12, 13, 15, 1742, 43, 45, 4792, 93, 95, 9712 outcomes in totala table lists every outcome systematically — nothing missed.
WE 2
Find P(event)
T is the event “the two-digit number is a multiple of 3”. Find P(T).
P(T) = n(T)/n(U)multiples of 3: 12, 15, 42, 45, 93 → n(T) = 5n(U) = 12P(T) = 5/12count the favourable outcomes, divide by the total.
WE 3
Use the complement
Using P(T) = 512 from WE 2, find P(T′).
P(T’) = 1 − P(T)= 1 − 5/12P(T’) = 7/12“not a multiple of 3” is just the complement.
WE 4
Use the union formula
A student has a dog with probability 0.8, a cat with probability 0.6, and a cat or a dog with probability 0.9. Find P(has both a dog and a cat).
P(D ∪ C) = P(D) + P(C) − P(D ∩ C)0.9 = 0.8 + 0.6 − P(D ∩ C)P(D ∩ C) = 1.4 − 0.9P(D ∩ C) = 0.5rearrange the union formula to get the intersection.
WE 5
Expected number of occurrences
The probability a spinner lands on red is 0.35. If it is spun 80 times, how many reds would you expect?
expected = np= 80 × 0.3528 reds (expected)it’s an average — you won’t get exactly 28 every time.
💡 Top tips
List the sample space (table or list) before counting — it stops you missing outcomes.
“At least one” or “not” → use the complement 1 − P.
∩ = and (overlap); ∪ = or (at least one). Learn the symbols cold.
Union formula subtracts the overlap so it isn’t counted twice.
Expected occurrences = np — quote it as an average.
A Venn or tree diagram can help even when the question doesn’t ask for one.
⚠ Common mistakes
Forgetting to subtract the overlap in the union formula — that double-counts A ∩ B.
Mixing up ∩ and ∪. ∩ is “and” (both); ∪ is “or” (either).
Treating np as exact. It’s the expected average, not a guaranteed count.
Missing outcomes when listing — use a systematic table.
Probabilities outside 0–1. Every probability must lie between 0 and 1.
Forgetting P(A) + P(A′) = 1 when the complement is the faster route.
Next up — Independent & Mutually Exclusive Events. You’ve met the union and intersection; next you’ll see two special cases that simplify them: mutually exclusive events (can’t both happen, so P(A ∩ B) = 0) and independent events (one doesn’t affect the other, so P(A ∩ B) = P(A)P(B)).
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