Lesson 15.3 · 20 min
Experimental Probability
What actually happens when you run a trial, how it compares with the theory, and how to predict a frequency.
In this lesson you will learn to…
- Calculate experimental probability from the results of a trial
- Explain why more trials give a more reliable estimate
- Decide from results whether a dice or spinner looks biased
- Calculate an expected frequency
Two different things called “probability”
There are two ways to put a number on how likely something is, and it matters which one you are doing.
Theoretical probability is worked out by reasoning. A fair dice has six equally likely faces, so . You never touch a dice to work this out.
Experimental probability is worked out by doing it. Roll the dice 60 times, count 12 sixes, and your estimate is . It is also called relative frequency.
Why experimental probability wobbles
Roll a fair dice six times and you will very rarely get exactly one six. That is not because the dice is broken — it is because six trials is a tiny sample.
The more trials you run, the less the wobble matters:
| Trials | A typical estimate of | How far from |
|---|---|---|
| 6 | or | miles off |
| 60 | around to | close |
| 600 | around to | very close |
Use the spinner above to see this for yourself. It is much more convincing when you watch it happen than when you read about it.
When you have no choice but to experiment
Theoretical probability needs equally likely outcomes. Plenty of real situations do not have any:
- Will a drawing pin land point up? The two ways it can land are not symmetrical.
- Will this bus be late? There is no list of equally likely outcomes.
- Will this particular biased dice show a 6? It is biased — that is the whole point.
For all of these, running trials is the only route to a number.
Is it biased?
To judge fairness, compare what happened with what a fair object would be expected to give — and then ask whether the gap is big enough to matter for that number of trials.
A gap of 10 out of 30 trials proves nothing. The same gap out of 3000 trials is damning. Always mention the number of trials when you give your verdict.
Predicting with expected frequency
The same formula works forwards. If a seed has a probability of germinating and you plant 250, you expect to come up.
Expected frequency is an estimate, not a promise. Saying “exactly 200 will germinate” is wrong; saying “about 200” is right.
Key rules and formulas
Experimental probability (relative frequency)
This is what you measure. It changes every time you repeat the experiment.Theoretical probability
This is what you calculate, assuming the dice or spinner is fair. It never changes.More trials, closer agreement
As the number of trials grows, the experimental probability settles down towards the theoretical one. Ten spins tells you almost nothing; a thousand tells you a great deal.Expected frequency
Use it to predict how often something should happen. It rarely comes out as a whole number — round sensibly and say it is an estimate.
How many spins is enough?
| Outcome | Red | Blue | Green | Yellow |
|---|---|---|---|---|
| Frequency | 0 | 0 | 0 | 0 |
| Relative frequency | — | — | — | — |
- Number of spins
- 0
- Experimental P(Red)
- —
- Theoretical P(Red)
- 1/4 = 0.25
- Difference
- —
Press +10 spins a few times and watch the relative frequency for Red jump around. Then press +100 spins twice and watch it settle near . That settling is the whole idea of this lesson.
Worked examples
Experimental probability from results
Try it yourself first, then reveal the solution one step at a time.
Is the dice biased?
Try it yourself first, then reveal the solution one step at a time.
Expected frequency
Try it yourself first, then reveal the solution one step at a time.
Practice: experimental probability
10 questions
Question 1 of 10Type the answerFoundation A coin is tossed 80 times and lands heads 46 times. Calculate the experimental probability of a head, as a decimal.Question 2 of 10Type the answerFoundation The probability a seed germinates is . A gardener plants 250 seeds. How many would you expect to germinate?Question 3 of 10Explore and match Drag the marker to show the probability of rolling an even number on a fair six-sided dice.Move the controls until you match the question, then press Check.
- Probability
- 0
Question 4 of 10Explore and matchFoundation Run the spinner at least 100 times using the buttons, and watch the relative frequency for Red as the number of spins grows.Move the controls until you match the question, then press Check.
Results so far Outcome Red Blue Green Yellow Frequency 0 0 0 0 Relative frequency — — — — - Number of spins
- 0
- Experimental P(Red)
- —
- Theoretical P(Red)
- 1/4 = 0.25
- Difference
- —
Question 5 of 10Multiple choice A dice is rolled 600 times and gives 145 sixes. Which conclusion is best supported?Question 6 of 10Match the pairs Match each term to its meaning.Pick an answer, then choose the row it belongs to. You can also drag it.
- Theoretical probability
- Experimental probability
- Expected frequency
- Biased
Question 7 of 10True or false If the experimental probability does not exactly match the theoretical probability, the object must be biased.Question 8 of 10Complete the table A four-sided spinner is spun 200 times. Complete the table of relative frequencies, as decimals.Colour Frequency Relative frequency Red 54 Blue 48 Green 62 Yellow 36 Question 9 of 10Flashcards Flip through the probability formulas until you can recall each one before turning the card.Card 1 of 7
Tap the card to flip it
Question 10 of 10Quick-fire drill Quick-fire: expected frequency. Each one is trials probability.8 questions · 90 seconds · answer as many as you can.