Matemáticas
Theoretical and Experimental Probability: What Separates Them
Theoretical probability is 1/6 from the die itself; experimental is 13/60 from what you rolled. A worked run from 30 to 1,200 rolls shows the gap closing.
Respuesta breve
What is the difference between theoretical and experimental probability?
Theoretical probability is worked out from the situation: one face in six, so 1/6. Experimental probability is counted from what actually happened: 13 sixes in 60 rolls, so 13/60.
La respuesta corta
Theoretical probability comes from reasoning about the situation: a fair die has six equally likely faces, so the probability of a six is 1/6. Experimental probability comes from counting what happened: 13 sixes in 60 rolls gives 13/60. More trials pull the experimental figure towards the theoretical one.
Qué cambia la respuesta
- Whether the die, coin or spinner is actually fair — if it is biased, only the experimental figure tells you anything true about it.
- How many trials were run: 20 rolls tell you almost nothing, and 600 tell you a great deal.
- Whether the outcomes are equally likely in the first place — theoretical probability needs that assumption, and a drawing pin landing point-up does not have it.
- What the question asks for: estimate the probability means use the experiment, while calculate or write down the probability means use the theory.
Two routes to the same kind of number
Theoretical probability is arithmetic done before anything happens. Count the outcomes you want, count all the outcomes there are, and divide — provided every outcome is equally likely. A fair die has one face showing six out of six faces, so P(6) = 1/6, and no amount of rolling is required to establish that.
Experimental probability, also called relative frequency, is arithmetic done after something has happened. Count how many times the event occurred, divide by the number of trials, and you have an estimate. Write it as a fraction, a decimal or a percentage — the exam accepts all three, but it will not accept a whole number of successes on its own.
The two answer slightly different questions. Theoretical probability answers what should happen to an ideal object. Experimental probability answers what did happen to a real one. That is the entire distinction, and everything else follows from it.
One die, counted both ways
The theoretical value is fixed at 1/6, which is 0.167 to three decimal places. Suppose a class shares out the rolling and records the running totals of sixes as they go. A set of results like the following is entirely ordinary:
Read down the last column and the point of the whole topic appears. The experimental value wanders early and settles late. Nothing forces it to improve at every stage — between 30 and 60 rolls above it actually gets worse — but over enough trials it closes in and stays close.
There is a second thing hidden in those numbers that most textbooks skip. The proportion gets closer to 1/6, but the raw count does not get closer to the expected number of sixes. Expected sixes at 30 rolls is 5 and the class got 3, out by 2. Expected at 1,200 is 200 and the class got 205, out by 5. The gap in the count grew while the gap in the proportion shrank. Long runs make the fraction reliable, not the tally.
- 30 rolls, 3 sixes — experimental probability 0.100, out by 0.067
- 60 rolls, 13 sixes — 0.217, out by 0.050
- 180 rolls, 34 sixes — 0.189, out by 0.022
- 600 rolls, 108 sixes — 0.180, out by 0.013
- 1,200 rolls, 205 sixes — 0.171, out by 0.004
How the gap is used to test for bias
Once a student can produce both numbers, the interesting question is what to do when they disagree. A small gap over few trials means nothing. A gap that persists over many hundreds of trials is evidence that the object is not fair — that is the reasoning behind every is this dice biased question on a paper.
The reverse calculation gets asked just as often. Expected frequency is probability multiplied by the number of trials: a fair die rolled 300 times gives an expected 300 × 1/6 = 50 sixes. If the same die is known to be biased with P(6) = 0.28, the expected number becomes 300 × 0.28 = 84. The mark is for multiplying, and for using the right probability of the two.
In our Year 8 and GCSE sessions we make students state which of the two they have used before they write the answer, because the arithmetic is almost never where marks are lost. Reaching for the theoretical value on a question about a bent spinner is.
Preguntas frecuentes
How many trials are enough for experimental probability?
There is no threshold that makes it correct, only more trials that make it more reliable. For classroom work, a few hundred is usually enough for the estimate to sit visibly near the theoretical value. Exam questions almost always hand you the number of trials and expect you to comment on whether it is large enough to trust.
Can experimental probability ever equal theoretical probability?
Yes, and it often does exactly. Roll a die 60 times, get 10 sixes, and both are 1/6. That agreement is a coincidence rather than a confirmation, though. A single matching result over 60 trials is far weaker evidence of fairness than a near-miss over 6,000.
Which one do I use if a question gives me both?
Read the object. If the question says fair, unbiased or ordinary, use the theoretical value even if trial data is printed alongside. If it says biased, bent, or gives you only a table of results, use the relative frequency from the table. The adjective in front of the noun decides it.
Why did my experiment give an answer nowhere near the theory?
Almost always because the number of trials was small. Twenty rolls of a fair die will quite often produce no sixes at all, which gives an experimental probability of 0 — a correct calculation from insufficient data. Pool results across the class before drawing any conclusion.
Última actualización
¿Sigues con dudas?
Cuéntanos la situación y te diremos con claridad qué haríamos, incluso si la respuesta es que no nos necesitas.