Probability scale, theoretical vs experimental
A probability is a number between 0 and 1 that measures how likely an event is. There are two main ways to assign probabilities — and they should match in the long run.
The 0-to-1 scale
- 0 = impossible.
- 1 = certain.
- 0.5 = "fifty-fifty".
- Anything in between is "less likely than not" (closer to 0) or "more likely than not" (closer to 1).
You'll be asked to mark events on a probability scale (a number line from 0 to 1) with arrows or letters.
Examples:
- "It will rain in London this year": close to 1.
- "A fair coin lands heads": exactly 0.5.
- "A randomly chosen day in February has 30 days": 0.
Probabilities are commonly written as fractions (3/8), decimals (0.375), or percentages (37.5%). Use the form the question asks for.
Theoretical (a priori) probability
If outcomes are equally likely, count favourable outcomes and divide by the total. We met this in P2.
P(rolling a 4 on a fair die) = 1/6 — calculated from the structure of the die.
Experimental (a posteriori) probability
Run trials, count occurrences, divide.
Relative frequency = (number of times A occurred) / (total trials).
This estimates PA. The more trials, the better the estimate (Law of Large Numbers).
Relative frequency vs theoretical — when do they agree?
For a fair object, experimental probability converges to theoretical probability as the number of trials grows. After only 10 spins of a fair coin, you might see 7 heads (0.7) — but after 10,000 spins you'd expect to see ≈ 5,000 heads (0.5).
For a biased object, theoretical probability isn't 1/n; only experimental data gives you PA. That's why we use surveys, simulations and large samples in real-world probability.
Predicting outcomes
Given PA (theoretical or estimated), the predicted number of times A occurs in N trials is N × PA.
Worked example: a four-sided spinner is spun 80 times. From earlier trials, P(red) is estimated as 0.45.
- Predicted reds = 80 × 0.45 = 36.
Comparing fairness
Given relative frequencies for two students rolling the same die:
- Student A: 60 rolls, 7 sixes → 7/60 ≈ 0.117.
- Student B: 600 rolls, 95 sixes → 95/600 ≈ 0.158.
Whose estimate of P(6) is more reliable? Student B, because they ran more trials. With 600 rolls the random fluctuation is much smaller.
⚠Common mistakes— Common mistakes (examiner traps)
- Probability > 1 or < 0. Always check.
- Stating "1 in 6" as a probability when the question wants a fraction. Write 1/6.
- Comparing relative frequencies from different sample sizes without weighting.
- Confusing "more reliable" with "closer to 1/2". Reliability depends on sample size, not how close to fair.
- Forgetting to multiply when asked for predicted counts.
➜Try this— Quick check
A spinner has P(blue) = 0.2 (theoretical). After 50 spins, blue comes up 16 times. (a) Compute the experimental probability of blue. (b) Is this evidence the spinner is unfair? Comment briefly.
Answers: (a) 16/50 = 0.32; (b) Higher than 0.2, but a 50-spin sample is small — chance variation could explain it. Need more trials.
AI-generated · claude-opus-4-7 · v3-deep-probability