Experimental Probability
Experimental probability, also called relative frequency, is the number of times an outcome happens divided by the number of trials. In NSW Year 8 maths (NESA outcome MA4‑PRO‑C‑01) it is compared with the theoretical probability expected when conditions are fair.
You will work with hundreds of trials, pool results from several groups, predict counts for a larger group such as Saturday shoppers or learner drivers, and estimate the colours in a bag or on a spinner you cannot see. You will also decide when a result is evidence that a die or spinner is biased, and choose a fair simulation, such as random digits for a netball shooter who scores 70% of her shots. It is part of Year 8 Probability.
Every question with a fully worked solution.
- Experimental Probability - Video - Experimental Probability and Examples Watch
Theory
The experimental probability, or relative frequency, is \(\dfrac{\text{frequency}}{\text{number of trials}}\). If juice was chosen \(92\) times in \(400\) sales, its relative frequency is \(0.23\). Many trials give a good estimate, which can predict how often an outcome will happen. This Year 8 maths page shows how.
Frequency is the number of times something happens. A trial is one go of the experiment.
Experimental probability (relative frequency):
or, from a frequency table:
Theoretical probability is what fair, unbiased conditions should give, worked out by counting outcomes. Experimental results are usually close to it, but rarely exactly equal.
All outcomes total 1. The relative frequencies of all the outcomes add to \(1\), so a missing one is found by subtraction: \(1 - 0.45 - 0.3 - 0.12 = 0.13\).
Predicting. Use the relative frequency as the probability: expected number \(=\) probability \(\times\) number of trials. A count of people or objects is a whole number, so round it.
More trials, better estimate. To combine groups, add all the frequencies and all the trials.
Bias and fairness. A result far from the theoretical probability over hundreds of trials suggests the die, coin or spinner is biased, or the game is unfair. Over a handful of trials it suggests nothing.
Simulation. Model the event with something that has the same probability. For \(P = 0.2\), use random digits \(0\) to \(9\) with \(0\) and \(1\) meaning the event happened (\(2\) of the \(10\) digits).
What to work out:
| Asked for | Work out |
|---|---|
| relative frequency | frequency \(\div\) number of trials |
| expected number of times | probability \(\times\) number of trials |
| a frequency | relative frequency \(\times\) number of trials |
| the number of trials | frequency \(\div\) relative frequency |
| how many of \(n\) objects | relative frequency \(\times\, n\), then round |
Using experimental probability
- Add all the frequencies to get the number of trials.
- Divide the frequency of the event by the number of trials.
- Compare with the theoretical probability if there is one.
- Predict: multiply the probability by the new number of trials, and round a count of things.
- Judge bias only from a large number of trials.
Running a simulation
- Choose a device with the same probability (random digits, a die, a coin, a spinner).
- Decide which results mean the event happened.
- Count the successes in the list and divide by the number of trials.
(i) a sum of \(10\)
(ii) an odd sum
(iii) a sum less than \(6\).
| Sum | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|
| Freq. | 0 | 2 | 4 | 4 | 5 | 7 |
| Sum | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|
| Freq. | 6 | 4 | 4 | 3 | 1 |
The frequencies add to \(40\), so there were \(40\) trials. Odd sums: \(2 + 4 + 7 + 4 + 3 = 20\). Sums less than \(6\): \(0 + 2 + 4 + 4 = 10\).
| \(P(10)\) | \(=\) | \(\dfrac{4}{40} = \dfrac{1}{10}\) |
| \(P(\text{odd})\) | \(=\) | \(\dfrac{20}{40} = \dfrac{1}{2}\) |
| \(P(\text{under } 6)\) | \(=\) | \(\dfrac{10}{40} = \dfrac{1}{4}\) |
(i) \(\dfrac{1}{10}\)
(ii) \(\dfrac{1}{2}\) (sums \(3, 5, 7, 9, 11\))
(iii) \(\dfrac{1}{4}\) (sums \(2, 3, 4, 5\))
(i) the relative frequency of juice, as a decimal and a percentage
(ii) the relative frequency of water or milk.
Trials: \(148 + 92 + 64 + 96 = 400\).
| \(\dfrac{92}{400}\) | \(=\) | \(\dfrac{23}{100} = 0.23\) |
| \(\dfrac{148 + 64}{400}\) | \(=\) | \(\dfrac{53}{100} = 0.53\) |
(i) \(0.23\), or \(23\%\)
(ii) \(0.53\)
(i) the theoretical probability
(ii) the expected number in \(500\) rolls
(iii) the experimental probability, as a decimal.
(iv) Is the die biased?
| \(P(7, 8, 9, 10)\) | \(=\) | \(\dfrac{4}{10} = 0.4\) |
| \(0.4 \times 500\) | \(=\) | \(200\) |
| \(\dfrac{189}{500}\) | \(=\) | \(\dfrac{378}{1000} = 0.378\) |
(i) \(0.4\)
(ii) \(200\)
(iii) \(0.378\)
(iv) A gap of \(11\) in \(500\) rolls is no evidence of bias.
(i) the relative frequency of arriving on time
(ii) how many to expect on time
(iii) how many to expect late.
| \(\dfrac{496}{640}\) | \(=\) | \(\dfrac{31}{40} = 0.775\) |
| \(0.775 \times 1150\) | \(=\) | \(891.25\) |
| \(1150 - 891\) | \(=\) | \(259\) |
(i) \(0.775\)
(ii) about \(891\) (services are whole numbers)
(iii) about \(259\)
(i) group 2’s relative frequency
(ii) the pooled relative frequency
(iii) the point-up count to expect in \(2000\) drops.
| \(\dfrac{66}{150}\) | \(=\) | \(\dfrac{44}{100} = 0.44\) |
| \(\dfrac{41 + 66 + 108}{100 + 150 + 250}\) | \(=\) | \(\dfrac{215}{500} = 0.43\) |
| \(0.43 \times 2000\) | \(=\) | \(860\) |
(i) \(0.44\)
(ii) \(0.43\), the better estimate because it uses all \(500\) drops
(iii) about \(860\)
(i) Find the relative frequency of blue, to \(2\) decimal places.
(ii) Estimate the number of blue counters.
(iii) Estimate the number of yellow counters.
| \(\dfrac{56}{120}\) | \(=\) | \(0.466\ldots \approx 0.47\) |
| \(0.47 \times 15\) | \(=\) | \(7.05 \approx 7\) |
| \(15 - 7\) | \(=\) | \(8\) |
(i) \(0.47\)
(ii) about \(7\) blue
(iii) about \(8\) yellow
\(4, 0, 7, 1, 9, 3, 6, 8, 2, 5, 1, 7, 0, 4, 6, 9, 3, 8, 5, 2\).
(i) Why do \(0\) and \(1\) model a save?
(ii) How many saves were simulated?
(iii) Find the relative frequency of a save.
| \(P(0 \text{ or } 1)\) | \(=\) | \(\dfrac{2}{10} = 0.2\) |
| \(\text{Saves}\) | \(=\) | \(4\) |
| \(\dfrac{4}{20}\) | \(=\) | \(0.2\) |
(i) Two of the ten digits is \(20\%\), the same as the save rate.
(ii) \(4\) saves (two \(0\)s and two \(1\)s)
(iii) \(0.2\)
Common pitfalls
Frequently asked questions
What is relative frequency?
The number of times an outcome happened divided by the number of trials. It is also called experimental probability.
How is experimental probability different from theoretical probability?
Experimental probability comes from the results of trials; theoretical probability comes from counting equally likely outcomes.
How do I predict how many times something will happen?
Multiply the probability (or relative frequency) by the number of trials, and round if you are counting people or objects.
How many trials do I need before I can say a die is biased?
Hundreds. A few trials can land far from the theoretical probability just by chance.
How do I combine results from several groups?
Add all the frequencies and all the trials, then divide. Do not average the separate relative frequencies.