Possibility spaces and combined experiments
A possibility space (or sample space) is the complete list of equally likely outcomes for an experiment. Building one carefully is the foundation of any probability calculation.
Single-experiment sample spaces
- A coin: {H, T} — 2 outcomes.
- A 6-sided die: {1, 2, 3, 4, 5, 6} — 6 outcomes.
- A spinner with 5 equal sectors labelled A–E: {A, B, C, D, E} — 5 outcomes.
- A pack of 52 cards: 52 outcomes.
For a fair experiment, each outcome has probability 1/n where n is the number of outcomes.
Combined experiments
If you do two (or more) experiments, the combined sample space contains every ordered combination.
Two coins: {HH, HT, TH, TT} — 4 outcomes (2 × 2). A coin and a die: 2 × 6 = 12 outcomes — H1, H2, …, H6, T1, …, T6. Two dice: 36 outcomes (6 × 6, see P6).
Use the product rule for counting (N5): if experiment 1 has m outcomes and experiment 2 has n outcomes, the combined experiment has m × n outcomes.
Building a sample space — methods
- List systematically for small cases: easier to spot patterns and avoid duplicates.
- Tree diagram: branches at each stage; each path is one outcome.
- Grid (table): ideal for two-stage experiments.
- Set notation: handy for shorthand (we'll see this in algebra-style probabilities).
✦Worked example— Worked example — two coin tosses
Outcomes: HH, HT, TH, TT (each with probability 1/4).
- P(at least one head) = 3/4.
- P(both same) = 2/4 = 1/2.
- P(exactly one head) = 2/4 = 1/2.
✦Worked example— Worked example — drawing balls from a bag
A bag has 3 red and 2 blue balls. Two are drawn without replacement. List the sample space (treat distinct balls).
Label the balls R₁ R₂ R₃ B₁ B₂. Combinations of 2 from 5: C(5,2) = 10 combinations.
- 3 red-red combinations.
- 1 blue-blue combination.
- 6 red-blue combinations.
P(both red) = 3/10. P(both blue) = 1/10. P(one of each) = 6/10 = 3/5.
(With replacement: 5 × 5 = 25 ordered outcomes; some calculations differ.)
With or without replacement
This is the critical distinction.
With replacement: each draw is independent; probabilities don't change. Without replacement: probabilities change because the bag's composition changes.
Worked example: a bag has 4 red and 6 blue balls.
- With replacement, P(red) on the second draw = 4/10 = 2/5 (unchanged).
- Without replacement, P(red on second | red on first) = 3/9 = 1/3.
Trees and stages
For multi-stage experiments, build a tree (P8 covers probability trees in detail). Each branch represents one stage; each path represents one full outcome.
Worked example: toss a coin then roll a die. The tree has 2 first-level branches × 6 second-level branches = 12 paths. P(H and 6) = 1/12.
⚠Common mistakes— Common mistakes (examiner traps)
- Listing pairs as unordered when order matters. (H, T) and (T, H) are usually distinct outcomes.
- Confusing "with replacement" and "without replacement" — they give different sample-space sizes and different probabilities.
- Treating identical-looking balls as the same outcome. When balls of the same colour are physically distinct, count them separately for equally likely sample spaces.
- Multiplying instead of listing for small cases. A 4-outcome example is fastest by listing, not algebra.
- Forgetting that the sample space gives equal weight to each outcome only if the experiments are fair.
➜Try this— Quick check
A 3-coin toss is performed. List the sample space and find P(exactly 2 heads).
Sample space: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT (8 outcomes). Exactly 2 heads: HHT, HTH, THH — 3 outcomes. P = 3/8.
AI-generated · claude-opus-4-7 · v3-deep-probability