Randomness, fairness and expected outcomes
Probability is the formal language of randomness. Before computing anything, decide whether the situation is fair (every outcome equally likely) or biased.
Equally likely outcomes — the classical definition
When all outcomes of an experiment are equally likely, the probability of an event A is:
P(A) = (number of outcomes in A) / (total number of outcomes)
This is the classical definition. It only works when each outcome really is equally likely — fair coin, fair die, well-shuffled deck, etc.
Worked example: rolling a fair 6-sided die. P(rolling a number > 4) = 2/6 = 1/3.
Fair vs biased
A fair coin lands H or T each with probability 0.5. A biased coin might give P(H) = 0.7. To decide fairness, look at:
- Symmetry of the object (a regular cube is geometrically fair).
- Empirical evidence (relative frequency from many trials).
If you suspect bias, the relative frequency method (P1) gives a reasonable estimate.
Expected number of outcomes
If an event has probability PA and the experiment is repeated N times, the expected number of times A occurs is:
E(A) = N × P(A)
Worked example: a fair die is rolled 60 times. Expected number of 6s?
- P(6) = 1/6.
- E = 60 × 1/6 = 10.
⚠ "Expected" doesn't mean "guaranteed". You might roll 8 sixes or 13 sixes; 10 is the long-run average.
Worked example with bias: a spinner has P(red) = 0.3. It is spun 50 times.
- Expected number of reds = 50 × 0.3 = 15.
Multiple outcomes — partition the space
The set of possible outcomes is the sample space. It must be:
- Exhaustive — covers every possibility.
- Mutually exclusive — no two events overlap.
The probabilities of all outcomes in a partition must sum to 1 (see P4).
Estimating from data
If a die has been rolled many times and outcomes recorded, the relative frequency converges to the true probability. Use this to:
- Estimate P(outcome) for biased objects.
- Predict expected counts in further trials.
⚠Common mistakes— Common mistakes (examiner traps)
- Assuming "fair" when the question gives data suggesting bias. Check before using 1/n.
- Confusing expected with guaranteed. Always say "expected" or "on average".
- Probabilities greater than 1. Often a unit-conversion mistake (e.g. miscounting outcomes).
- Not multiplying by N for expected counts in many trials.
- Treating "at least one 6 in two rolls" as 2 × 1/6 = 1/3. This is wrong — see P8 for combined events.
➜Try this— Quick check
A bag holds 12 balls: 5 red, 4 green, 3 blue. A ball is drawn at random. (a) P(red). (b) Expected number of greens in 60 draws (with replacement).
Answers: (a) 5/12; (b) 60 × 4/12 = 20.
AI-generated · claude-opus-4-7 · v3-deep-probability