Experimental vs Theoretical Probability

In the last lesson you worked out probabilities by counting equally likely outcomes — this is theoretical probability. But many real situations cannot be analysed that way: a drawing pin, a bent coin or a hand-made spinner has no guarantee that its outcomes are equally likely. In those cases we collect data and use experimental probability, also called relative frequency. This lesson explains how to estimate a probability from experiment, how it compares with theory, what "fair" really means, and how more trials give a better estimate.

This is core AO1 (calculating relative frequency) and AO2/AO3 content (comparing observed and expected results, and deciding whether an object is fair). OCR loves to ask you to calculate a relative frequency, then give a reason about fairness, so precise reasoning earns marks here.

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Key Vocabulary

Term	Definition
Theoretical probability	Probability worked out from equally likely outcomes, e.g. a fair dice has $P(6) = \dfrac{1}{6}$P(6)=61​.
Experimental probability	Probability estimated from the results of an experiment.
Relative frequency	Another name for experimental probability: $\dfrac{\text{frequency of event}}{\text{total trials}}$total trialsfrequency of event​.
Trial	One repetition of an experiment.
Frequency	The number of times an outcome actually occurs.
Expected frequency	The number of times an outcome should occur in theory: $P \times n$P×n.
Fair	All outcomes equally likely (no bias).
Biased	Outcomes are not equally likely; the object favours certain outcomes.

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Relative Frequency (Experimental Probability)

When outcomes may not be equally likely, we do the experiment many times and use

$\text{relative frequency} = \dfrac{\text{number of times the event occurs}}{\text{total number of trials}}$relative frequency=total number of trialsnumber of times the event occurs​

This is our best estimate of the true probability based on the data we have. The crucial word is estimate: a relative frequency is not the exact probability, it is a number we have measured from a sample. If you repeated the same experiment, you would almost certainly get a slightly different relative frequency, because experiments contain natural random variation. The larger the sample, the smaller this variation tends to be, which is why a relative frequency from thousands of trials is trusted far more than one from a handful.

Relative frequency is the only tool available when we cannot assume equally likely outcomes. A drawing pin, a thumbtack, a real-world weather event, a manufactured component or a sports result all fall into this category — there is no symmetry that makes the outcomes equally likely, so the theoretical formula simply does not apply. In every such case we collect data and divide, exactly as in the formula above.

Worked Example 1

A drawing pin is dropped $200$200 times. It lands "point up" $72$72 times. Work out the relative frequency of landing point up.

Solution: $\text{relative frequency} = \dfrac{72}{200} = 0.36$relative frequency=20072​=0.36

Common error: Calling this "the probability". It is an estimate of the probability; with different trials you would get a slightly different value.

Worked Example 2

A spinner is spun $50$50 times and lands on red $19$19 times. Work out the relative frequency of red as a decimal.

Solution: $\dfrac{19}{50} = 0.38$5019​=0.38

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Theoretical vs Experimental: Two Different Routes

For a fair object you can calculate the theoretical probability directly. For any object you can estimate it experimentally. The two should be close if the object is fair — but they will rarely be exactly equal, because experiments contain natural variation.

Worked Example 3

A fair coin is tossed $40$40 times and lands on heads $24$24 times. (a) Write down the theoretical probability of heads. (b) Work out the experimental probability (relative frequency) of heads. (c) Give a reason why the two values are different.

Solution: (a) For a fair coin, $P(\text{heads}) = \dfrac{1}{2} = 0.5$P(heads)=21​=0.5. (b) Relative frequency $= \dfrac{24}{40} = 0.6$=4024​=0.6. (c) The difference is due to natural random variation; $40$40 tosses is a small number, so the experimental value will not match the theoretical value exactly.

Worked Example 4

A fair six-sided dice is rolled $300$300 times, giving these frequencies.

Score	1	2	3	4	5	6
Frequency	48	51	52	47	50	52

Show that there is little evidence the dice is biased.

Solution: For a fair dice each theoretical probability is $\dfrac{1}{6}$61​, so the expected frequency of each score is

$300 \times \dfrac{1}{6} = 50$300×61​=50

Every observed frequency ($48, 51, 52, 47, 50, 52$48,51,52,47,50,52) is within $3$3 of $50$50. These are small differences consistent with random variation, so there is little evidence of bias.

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Deciding Whether Something Is Fair

An object is biased if its outcomes are not equally likely. You test for bias by comparing the relative frequency with the theoretical probability (for a fair object) and looking for a large, consistent difference. The word "consistent" matters: a single odd result from a small sample proves nothing, because even a perfectly fair coin will occasionally produce a run of heads. What gives real evidence is a clear, repeatable gap between the observed relative frequency and the value you would expect if the object were fair, ideally backed up by a large number of trials.

This is why your language is assessed as well as your arithmetic. You can almost never say "the dice is biased" with certainty from one experiment, because the difference might be down to chance. The phrasing examiners reward is cautious — "there is evidence to suggest the dice is biased" — and the surest way to strengthen any such conclusion is to repeat the experiment with more trials.

Worked Example 5

A coin is tossed $200$200 times and lands on heads $130$130 times. (a) Work out the relative frequency of heads. (b) Is there evidence the coin is biased? Give a reason.

Solution: (a) $\dfrac{130}{200} = 0.65$200130​=0.65. (b) A fair coin would give a relative frequency near $0.5$0.5, so about $100$100 heads expected. Getting $130$130 is well above this, so there is evidence the coin is biased towards heads.

Common error: Saying "the coin is biased". We can never be certain from one experiment; the correct phrasing is "there is evidence to suggest the coin is biased".

Worked Example 6

A four-sided spinner is spun $400$400 times.

Score	1	2	3	4
Frequency	95	102	98	105

Is the spinner biased? Give a reason for your answer.

Solution: A fair four-sided spinner has expected frequency $400 \div 4 = 100$400÷4=100 for each score. The observed values ($95, 102, 98, 105$95,102,98,105) are all close to $100$100, so there is no strong evidence of bias.

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Expected vs Observed Frequency

The expected frequency of an event is what theory predicts:

$\text{expected frequency} = P(\text{event}) \times \text{number of trials}$expected frequency=P(event)×number of trials

Comparing the observed frequency (what actually happened) with the expected frequency is the heart of testing for fairness.

Worked Example 7

A fair dice is rolled $120$120 times. Work out the expected frequency of rolling a number greater than $4$4.

Solution: Numbers greater than $4$4 are $\{5, 6\}${5,6}, so $P = \dfrac{2}{6} = \dfrac{1}{3}$P=62​=31​.

$\text{expected frequency} = \dfrac{1}{3} \times 120 = 40$expected frequency=31​×120=40

Worked Example 8

A bag is known to contain red and blue counters in the ratio $3 : 2$3:2. A counter is drawn, its colour noted, and replaced, $250$250 times. Work out the expected number of red draws.

Solution: The ratio $3 : 2$3:2 gives $P(\text{red}) = \dfrac{3}{5}$P(red)=53​.

$\text{expected red} = \dfrac{3}{5} \times 250 = 150$expected red=53​×250=150

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More Trials Give a Better Estimate

As the number of trials increases, the relative frequency settles down and gets closer to the true probability. This is why a large experiment is more trustworthy than a small one. With only a handful of trials the relative frequency can swing wildly — three heads from four tosses gives $0.75$0.75, nowhere near $0.5$0.5 — but as the trials pile up, the early swings are swamped by the sheer weight of data and the value steadies. This behaviour is so reliable that it has a name, the law of large numbers, and it is the reason opinion pollsters, insurers and quality-control engineers all insist on large samples before trusting an estimate.