Experimental and Theoretical Probability

This lesson covers the difference between experimental (empirical) probability and theoretical probability, as required by the AQA GCSE Mathematics specification. You will learn how to estimate probabilities from experiments, understand relative frequency, and determine whether an object (such as a die or spinner) is fair or biased.

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Theoretical Probability

Theoretical probability is calculated by reasoning about equally likely outcomes, without carrying out an experiment.

P(event) = number of favourable outcomes / total number of equally likely outcomes

For example, the theoretical probability of rolling a 3 on a fair die is 1/6, because there are 6 equally likely outcomes and exactly one of them is a 3.

Theoretical probability assumes that the situation is fair (unbiased). If a coin is fair, P(heads) = 1/2 exactly.

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Experimental Probability (Relative Frequency)

Experimental probability (also called relative frequency) is calculated from the results of an actual experiment or from observed data.

Relative frequency = number of times the event occurred / total number of trials

Worked Example 1

A spinner is spun 50 times. The results are shown below.

Colour	Frequency
Red	18
Blue	12
Green	20
Total	50

The experimental probability (relative frequency) of landing on blue is:

P(blue) = 12/50 = 6/25 = 0.24

Exam Tip: Relative frequency is only an estimate of the true probability. It is not exact, but it becomes more reliable as the number of trials increases.

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Comparing Experimental and Theoretical Probability

If a spinner has three equal sections (red, blue, green), the theoretical probability of each colour is 1/3 (approximately 0.333).

Compare this with the experimental results above:

Colour	Theoretical Probability	Experimental Probability
Red	1/3 = 0.333	18/50 = 0.36
Blue	1/3 = 0.333	12/50 = 0.24
Green	1/3 = 0.333	20/50 = 0.40

The experimental probabilities are close to the theoretical values but not exactly equal. This is normal — random variation means that short experiments rarely match theory perfectly.

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The Law of Large Numbers

The law of large numbers states that as the number of trials increases, the experimental probability (relative frequency) tends to get closer to the theoretical probability.

graph LR
    A[Small number of trials] -->|Results vary widely| B[Large gap between experimental and theoretical probability]
    C[Large number of trials] -->|Results stabilise| D[Experimental probability close to theoretical probability]

Worked Example 2

A coin is flipped repeatedly. The cumulative relative frequency of heads is recorded:

Number of flips	Number of heads	Relative frequency
10	7	0.70
50	28	0.56
100	53	0.53
500	258	0.516
1000	507	0.507
5000	2512	0.502

Notice how the relative frequency settles closer to 0.5 (the theoretical probability for a fair coin) as the number of flips increases.

Exam Tip: If you are asked whether a coin or die is fair, compare the relative frequency to the theoretical probability. If the results are close after a large number of trials, the object is likely fair. If they differ significantly, it may be biased.

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Fair or Biased?

An object is fair if all outcomes are equally likely. An object is biased if some outcomes are more likely than others.

How to Decide if an Object is Biased

1. Carry out a large number of trials (the more the better).
2. Calculate the relative frequency of each outcome.
3. Compare with the theoretical probability for a fair object.
4. If there is a significant and persistent difference, the object may be biased.

Worked Example 3

A die is rolled 600 times. The results are:

Score	Frequency	Relative Frequency	Expected if Fair
1	95	0.158	100
2	102	0.170	100
3	98	0.163	100
4	105	0.175	100
5	97	0.162	100
6	103	0.172	100

All relative frequencies are close to 1/6 (approximately 0.167). This suggests the die is fair.

Worked Example 4

A different die is rolled 600 times:

Score	Frequency	Relative Frequency	Expected if Fair
1	70	0.117	100
2	75	0.125	100
3	80	0.133	100
4	85	0.142	100
5	110	0.183	100
6	180	0.300	100

The score of 6 appears much more often than expected. After 600 trials, this is a significant difference, suggesting the die is biased towards 6.

Exam Tip: Never say a die is definitely biased or definitely fair based on a small experiment. Use cautious language: "The results suggest the die may be biased" or "There is not enough evidence to say the die is biased — more trials are needed."

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Using Relative Frequency to Estimate Probability

When outcomes are not equally likely (for example, an irregularly shaped object), you cannot use theoretical probability. Instead, you must rely on experimental data.

Worked Example 5

A drawing pin is dropped 200 times. It lands "point up" 68 times.

Estimate the probability that the drawing pin lands point up:

P(point up) = 68/200 = 17/50 = 0.34

This estimate will be more reliable if more trials are carried out.

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Summary