Experimental Probability

This lesson covers experimental probability (relative frequency), estimating expected outcomes from experiments, comparing theoretical and experimental probability, and identifying bias. These topics are part of the Edexcel GCSE Mathematics (1MA1) specification and can appear on any of the three papers.

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Relative Frequency

Relative frequency is the proportion of times an event occurs in an experiment. It is an estimate of the probability based on collected data rather than equally likely outcomes.

Relative frequency of event A = number of times A occurs / total number of trials

Relative frequency is also called experimental probability or estimated probability.

Worked Example 1

A drawing pin is dropped 200 times. It lands "point up" 72 times. Estimate the probability that the drawing pin lands point up.

Solution: Relative frequency = 72/200 = 0.36

Key Point: Relative frequency is an estimate of probability, not the exact probability. The more trials you carry out, the more reliable the estimate becomes.

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Key Vocabulary

Term	Definition
Theoretical probability	Probability calculated from equally likely outcomes (e.g. fair dice: P(6) = 1/6)
Experimental probability	Probability estimated from the results of an experiment
Relative frequency	Another name for experimental probability
Trial	A single repetition of an experiment
Bias	When outcomes are not equally likely; the object favours certain outcomes
Fair	All outcomes are equally likely (no bias)

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Expected Outcomes

If you know (or estimate) a probability, you can predict how many times an event is expected to happen over a given number of trials:

Expected number of outcomes = probability × number of trials

Worked Example 2

A biased coin has P(heads) = 0.6. The coin is flipped 250 times. How many heads would you expect?

Solution: Expected heads = 0.6 × 250 = 150

Worked Example 3

A spinner is spun 80 times with the following results:

Colour	Red	Blue	Green	Yellow
Frequency	22	18	30	10

(a) Find the relative frequency of green. (b) The spinner is spun 500 more times. Estimate the number of times it will land on green.

Solution:

(a) Relative frequency of green = 30/80 = 3/8 (or 0.375)

(b) Expected green = 3/8 × 500 = 187.5

So you would expect approximately 187 or 188 greens.

Edexcel Exam Tip: Edexcel often asks you to use relative frequency to estimate the expected number of outcomes in a larger experiment. Show the multiplication clearly. If the answer is not a whole number, it is acceptable to leave it as a decimal or round to the nearest whole number — but state "approximately" or "about".

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Comparing Theoretical and Experimental Probability

For a fair object, we can calculate the theoretical probability. If the experimental results are very different from the theoretical prediction, this may suggest the object is biased.

Worked Example 4

A six-sided dice is rolled 300 times. The results are shown below.

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

Is there evidence that the dice is biased?

Solution: If the dice is fair, the theoretical probability of each score is 1/6. Expected frequency of each score = 300 × 1/6 = 50.

The observed frequencies (48, 51, 52, 47, 50, 52) are all very close to 50. There is no strong evidence that the dice is biased — the small variations are expected due to random chance.

Worked Example 5

A coin is tossed 200 times and lands on heads 130 times.

(a) Find the relative frequency of heads. (b) Comment on whether the coin is fair.

Solution:

(a) Relative frequency of heads = 130/200 = 0.65

(b) For a fair coin, P(heads) = 0.5, so in 200 tosses we would expect about 100 heads. Getting 130 heads is significantly more than expected. This suggests the coin may be biased towards heads.

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The Effect of Increasing the Number of Trials

As the number of trials increases, the relative frequency tends to get closer to the true (theoretical) probability. This is a fundamental principle of probability.

Illustration

Imagine tossing a fair coin:

• After 10 tosses, you might get 7 heads (relative frequency = 0.7)
• After 100 tosses, you might get 53 heads (relative frequency = 0.53)
• After 1000 tosses, you might get 508 heads (relative frequency = 0.508)
• After 10000 tosses, you might get 5012 heads (relative frequency = 0.5012)

The relative frequency converges towards 0.5.

Edexcel Exam Tip: If asked "How could you improve the reliability of the estimated probability?", the answer is always: carry out more trials. This is worth 1 mark and appears regularly.

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Identifying Bias

An object is biased (or unfair) if the outcomes are not equally likely.

You can identify bias by:

1. Comparing experimental results to theoretical predictions.
2. Looking for outcomes that occur much more or much less often than expected.
3. Carrying out a large number of trials to make the comparison reliable.

Worked Example 6

A four-sided spinner is labelled 1, 2, 3, 4. It is spun 400 times with the following results:

Score	1	2	3	4
Frequency	95	102	98	105

Is the spinner biased?

Solution: If the spinner is fair, the expected frequency for each score = 400 ÷ 4 = 100.

The observed frequencies (95, 102, 98, 105) are all close to 100. The differences are small and could be due to chance. There is no strong evidence that the spinner is biased.

Worked Example 7

The same spinner is spun another 400 times (800 total) and gives:

Score	1	2	3	4
Frequency	190	205	195	210

Expected frequency (if fair) = 800 ÷ 4 = 200.

The results are still reasonably close to 200. The variation is within what we might expect from random chance. More trials would give even more reliable evidence.

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Using Relative Frequency in Context

Worked Example 8

A factory tests light bulbs. Out of 5000 bulbs tested, 45 are defective.

(a) Estimate the probability that a randomly chosen bulb is defective. (b) The factory produces 80,000 bulbs. Estimate how many will be defective.

Solution:

(a) P(defective) = 45/5000 = 9/1000 (or 0.009)

(b) Expected defective = 9/1000 × 80,000 = 720 bulbs

Worked Example 9

A biased dice is rolled 60 times and lands on 6 a total of 18 times. Later it is rolled another 200 times. Estimate the number of 6s in the second batch.

Solution: Relative frequency of 6 = 18/60 = 3/10 = 0.3

Expected 6s in 200 rolls = 0.3 × 200 = 60 sixes

Edexcel Exam Tip: Use the relative frequency from the first experiment to estimate a count for the second. Show the multiplication as a single line of working.

Worked Example 10

A spinner has three colours: red, green and blue. In a trial of 500 spins, the results are:

Colour	Red	Green	Blue
Frequency	210	165	125