Theoretical probability is worked out from the equally likely outcomes, before any trial. Experimental probability is measured from trials that actually happened. They describe the same event from two directions, and they are allowed to disagree.
What is the small gap?
Students treat theory as a promise. A coin is “50-50”, so ten tosses must give five heads. The theoretical value is the long-run pattern, not a prediction for a short run.
How do the two compare?
| Theoretical | Experimental | |
|---|---|---|
| Source | list of equally likely outcomes | recorded trials |
| Formula | favourable ÷ total outcomes | times it happened ÷ trials |
| Changes with trials? | no | yes |
A scaffolded example
An original problem: a spinner has 5 equal sections, 2 coloured red. A student spins it 40 times and gets red 13 times.
Theoretical: 2 red out of 5 equal sections gives 2/5 = 0.4.
Experimental: 13 red out of 40 spins gives 13/40 = 0.325.
The gap is 0.4 − 0.325 = 0.075. With 40 spins, the expected red count is 0.4 × 40 = 16, and 13 is 3 below it. Nothing is wrong: a difference of 3 in 40 spins is ordinary.
If the student went on to 400 spins, the experimental value would tend to move nearer 0.4, but it would still not be exactly 0.4 every time.
What does the common mistake look like?
A student concludes from 13 reds in 40 spins that “the spinner is faulty”. A single short run cannot support that. The claim needs a much larger number of trials that remain far from 0.4.
The opposite slip is to use 13/40 when the question says the spinner is fair and asks for the probability. Then the theoretical value is wanted.
Self-check
A fair die is rolled 60 times and a 6 appears 7 times. Give the theoretical and experimental probabilities of a 6, and how many 6s you would expect.
Answer
Theoretical: 1/6 ≈ 0.167. Experimental: 7/60 ≈ 0.117.
Expected number: 60 × 1/6 = 10. The actual count, 7, is lower, but a difference of 3 in 60 rolls is unremarkable.
Where does this show up in SPM?
In SPM Mathematics, questions give a frequency table of trials and ask you to estimate a probability, or give a fair device and ask for the theory. In Science, genetics ratios are theory, and counted offspring are experiment.
The habit of asking “does this evidence support that claim?” appears in checking whether evidence supports a claim. Earlier steps are listing outcomes systematically and the hub.
One-to-one Mathematics tuition starts with a one-hour trial class (from RM50).