Experimental Probability
Experimental probability turns real results into estimates: you repeat a trial, record what happens, and use relative frequency to estimate how likely an outcome is. On this page you will learn how to find relative frequency from a results table, calculate expected frequency, and decide when an experimental estimate can be trusted, exactly as it is tested in Edexcel IGCSE Maths. When you are ready, scroll down to the auto-marked practice questions and put each skill to work.
What is Experimental Probability?
Sometimes you cannot work a probability out by pure reasoning. A drawing pin is not a fair coin: nobody can say in advance how likely it is to land point up. The only way to estimate that probability is to run an experiment: repeat the trial many times, count how often the outcome happens, and divide. The value you get is called the relative frequency, and it is your experimental probability.
To find a relative frequency from a results table:
- Find the total number of trials. If it is not given, add up all the frequencies in the table.
- Read off how many times your outcome happened. For an event like "an even number", add the frequencies of every score that counts.
- Divide the outcome count by the total. Leave the answer as a fraction, or as a decimal if it divides exactly.
A relative frequency is a probability, so it always sits between 0 and 1. If your answer is bigger than 1, you have divided the wrong way round.
Four Ideas to Master
The probability estimated from results. Read the counts from the table, then divide.
Probability of the event multiplied by the number of trials: how many times you expect it to happen.
Pool every set of results into one big experiment. More data always beats less.
For fair equipment you can also reason the probability out. Compare it with the experiment.
Why More Trials Give a Better Estimate
Here is a real simulated experiment. A spinner has 10 equal sections and 3 of them are red, so the true probability of red is 0.3. We spun it 500 times, and after every few spins we plotted the relative frequency of red so far.
Relative frequency of red, plotted against the number of spins
Across the page: number of spins (0 to 500). Up the side: relative frequency of red (0 to 1). The dashed orange line is the true probability, 0.3.
After 10 spins the relative frequency was 0.6, double the true value. After 100 spins it had dropped to 0.30, and from about 350 spins onwards it stayed within one hundredth of the truth. That is the key fact the examiners want you to state: the more trials you carry out, the closer the relative frequency gets to the true probability. A small experiment can mislead; a large one settles down.
Expected Frequency
Expected frequency turns a probability back into a count. If you know how likely one trial is to succeed, multiplying by the number of trials tells you how many successes to expect in total.
For example, a fair spinner has five equal sections and exactly one of them is gold, so each spin lands on gold with probability \(\dfrac{1}{5}\). In 45 spins:
\[ \begin{array}{rcl} \text{Expected golds} &=& \dfrac{1}{5} \times 45 \\[4pt] &=& 9 \end{array} \]Treat 9 as a prediction, not a promise: a real run of 45 spins will land either side of it, but no other single number is a better call. Watch for questions that ask about the event not happening: subtract the probability from 1 first, then multiply.
Experimental or Theoretical: Which Do You Trust?
Theoretical probability comes from reasoning about equally likely outcomes: a fair die gives each score a probability of \(\dfrac{1}{6}\) without a single roll. Experimental probability comes from data. Use these rules to answer comparison questions:
- If the equipment is fair and the outcomes are equally likely, the theoretical value is exact; the experiment only estimates it.
- If nothing says the equipment is fair (a drawing pin, a biased die, a bent coin), the experiment is all you have, so the relative frequency is the best available estimate.
- When two people test the same thing, the one with more trials has the more reliable estimate.
- For the best single estimate, combine every set of results: divide the total number of successes by the total number of trials.
- A big gap between the relative frequency and the theoretical value, after a large number of trials, is evidence the equipment may be biased.
Worked Examples
💡 Example 1: Relative Frequency from a Table
Nadia spins a five-colour spinner 40 times. The table shows her results. Find the relative frequency of blue, as a fraction and as a decimal.
| Colour | red | blue | green | yellow | purple |
|---|---|---|---|---|---|
| Frequency | 6 | 10 | 9 | 8 | 7 |
What's happening?
Blue came up 10 times out of 40 spins, so divide 10 by 40.
The frequencies already sum to 40, which confirms the total number of trials.
Both \(\dfrac{1}{4}\) and 0.25 earn the mark; they are the same number.
💡 Example 2: Expected Frequency
The probability that a biased coin lands heads is \(\dfrac{2}{5}\). Kofi flips it 60 times. How many heads should he expect?
What's happening?
Multiply the probability by the number of trials.
Two fifths of 60 is 24, so around 24 of the 60 flips should be heads.
The real count will vary, but 24 is the most likely value, and it is the answer the exam wants.
💡 Example 3: Expecting the Opposite
The probability that Priya's bus is late on a school morning is 0.15. Over the next 40 school mornings, on how many should she expect the bus NOT to be late?
What's happening?
The question asks about the bus not being late, so first subtract from 1 to get the complement.
Then it is a normal expected-frequency calculation: probability times trials.
Jumping straight to \(0.15 \times 40 = 6\) answers the wrong question.
💡 Example 4: The Best Single Estimate
Ben and Zola both test the same biased die by counting sixes. Whose estimate of the probability of a six is more reliable, and what is the best single estimate using all the results?
| Name | Rolls | Sixes |
|---|---|---|
| Ben | 30 | 9 |
| Zola | 120 | 32 |
Zola's estimate is more reliable: 120 rolls beat 30.
\[ \begin{array}{rcl} \text{Best estimate} &=& \dfrac{9 + 32}{30 + 120} \\[6pt] &=& \dfrac{41}{150} \end{array} \]What's happening?
More trials means a more reliable estimate, so on their own results Zola wins.
But the best single estimate uses everything: pool the successes (9 + 32) over the pooled trials (30 + 120).
Never average the two fractions; combine the raw counts instead.
🔑 Key Points
- Relative frequency \(= \dfrac{\text{number of times the outcome happens}}{\text{total number of trials}}\); it is the experimental probability.
- Every probability, experimental or theoretical, sits between 0 and 1.
- Expected frequency \(=\) probability \(\times\) number of trials.
- More trials pull the relative frequency towards the true probability.
- The best single estimate pools all successes over all trials.
⚠️ Common Pitfalls
- Dividing by the number of different outcomes instead of the number of trials.
- Missing the word not: subtract from 1 before multiplying for a complement.
- Trusting a 10-trial estimate over a 200-trial estimate because it "looks nicer".
- Averaging two relative frequencies instead of combining the raw counts.
- Expecting an experiment to match the theoretical value exactly; it only gets close.