Experimental Probability and Expected Frequency
Theoretical probability starts with a model. Experimental probability starts with observed results. Use both to make predictions, while remembering that a prediction is not a guarantee.
Foundation & Higher. Begin with Probability Basics and Rules. Multiplying a probability by a number of trials is an application of Direct Proportion.
Record results, then calculate relative frequency
A frequency is a count. The relative frequency of an outcome is its frequency divided by the total number of trials. It gives an estimate of the probability when the trials are suitable for the situation you want to predict.
| Spinner result | Frequency |
|---|---|
| Red | 36 |
| Blue | 27 |
| Green | 17 |
| Total | 80 |
Worked example — estimate from data
These are the results of 80 spins. Estimated P(red) = 36/80 = 9/20 = 0.45. Estimated P(not red) = 44/80 = 0.55.
The estimates total 1 because every spin has been recorded in exactly one category. Unlike a theoretical fair-spinner calculation, the estimate uses the observed counts, not the sizes of the sectors.
More trials and a more reliable estimate
For repeated trials under the same conditions, unbiased experimental proportions tend to settle near the model’s theoretical probabilities as the number of trials becomes large. A larger sample usually gives a more reliable estimate, but it does not guarantee a closer result at every stage.
| Trials | Heads | Chance |
|---|---|---|
| 10 | 7 | 0.70 |
| 50 | 28 | 0.56 |
| 200 | 103 | 0.515 |
A fair coin has theoretical P(heads) = 0.5. These cumulative results are consistent with the estimate settling towards 0.5. They do not prove the coin is fair, and the next result need not move the estimate closer.
More data do not repair systematic bias. Selecting only convenient trials, changing the spinner, or recording some results incorrectly can make a large experiment misleading. Use the same relevant conditions and record all outcomes.
Expected frequency: predict a number of outcomes
Expected frequency = number of trials × probability. Each suitable trial contributes the same chance to the predicted count. If you use an experimental estimate, the predicted frequency is also an estimate.
Worked example — use experimental probability
Using estimated P(red) = 0.45, predict the red results in 240 further spins under the same conditions. Expected red results = 240 × 0.45 = 108.
This predicts red results in the 240 new spins; it does not include the 36 red results already recorded. The actual count may be different.
Worked example — use a theoretical probability
For a fair six-sided die, P(6) = 1/6. In 90 rolls, the expected number of sixes is 90 × 1/6 = 15. This is the model prediction, not a promised count.
An expected count can be non-integer: 25 trials with success probability 0.3 give an expectation of 7.5. An actual count is a whole number. Retain 7.5 if asked for the expected value; if asked for a practical whole-number estimate, say about 8 and make the rounding clear.
Work backwards from an expected count
Rearrange the same relationship rather than inventing a new rule. If the expected count is E in n trials, the probability used is E ÷ n.
Worked example — find the probability or number of trials
A model predicts 42 successes in 150 trials. Its success probability is 42 ÷ 150 = 0.28.
At probability 0.28, an expected 70 successes would require 70 ÷ 0.28 = 250 trials. An observed 42 out of 150 would instead give an experimental estimate, not prove the true probability is 0.28.
Frequency trees organise counts
A frequency tree splits a total into categories and then subcategories. Each pair of child counts must add to its parent count. These labels are counts, not branch probabilities.
Here 48 + 32 = 80; 30 + 18 = 48; and 8 + 24 = 32. There are 38 wins altogether and 42 losses.
Worked example — read the relevant counts
Choose one recorded trial uniformly at random. P(red and win) = 30/80 = 3/8. P(win) = (30 + 8)/80 = 19/40.
To predict wins in 200 comparable future trials, use the observed win proportion: 200 × 19/40 = 95. Here the denominator 80 includes all recorded trials. Restricting the choice to just one branch is conditional probability, taught later.
Assumptions and the next trial
A theoretical model may assume a fair device or equally likely objects. A prediction from data assumes the observations are representative of future trials and the conditions remain relevant.
For independent fair coin tosses, five heads in a row do not make tails “due”. The next toss still has P(tails) = 1/2. Long-run proportions do not force a short run to balance itself.
Think Like a Mathematician
Two students estimate a success probability as 8/10 and 80/100. Both estimates are 0.8, but the second is based on more evidence. If both experiments are unbiased and comparable, which estimate would you usually trust more—and why does sample size alone not settle the question?
Common Mistakes
Using the wrong total: divide by all relevant recorded trials, not by the count of one other outcome.
Calling an expectation a guarantee: actual counts fluctuate.
Assuming a larger sample fixes bias: fair selection and consistent conditions still matter.
Practice
1. A spinner lands on yellow 18 times in 60 spins. Estimate P(yellow).
18/60 = 3/10 = 0.3.
2. Use that estimate to predict yellow results in 250 further comparable spins.
250 × 0.3 = 75 yellow results, as an estimate.
3. A fair die is rolled 120 times. Find the expected number of results greater than 4.
P(5 or 6) = 2/6 = 1/3, so 120 × 1/3 = 40.
4. An event has probability 0.35. Find its expected frequency in 30 trials. Must the answer be a whole number?
30 × 0.35 = 10.5. The expectation can be non-integer even though an actual count must be whole.
5. A model predicts 24 successes in 80 trials. What probability does it use?
24 ÷ 80 = 0.3.
6. In a frequency tree, 90 trials split into A:54 and B:36. A contains 21 wins; B contains 15 wins. Find both loss counts and the overall relative frequency of a win.
A losses = 54 − 21 = 33; B losses = 36 − 15 = 21. Win proportion = (21 + 15)/90 = 2/5. The four end counts 21 + 33 + 15 + 21 total 90.
7. A fair coin has produced six tails in succession. What is P(heads) on the next independent toss?
1/2. Previous results do not alter the model for an independent fair toss.
8. An experiment grows from 50 to 500 trials. Does its relative frequency have to become closer to the theoretical probability?
No. More comparable unbiased trials usually make the estimate more reliable, but a particular increase can move it further away. Bias can also persist.
Frequently Asked Questions
Are relative frequency and probability the same?
Relative frequency is an observed proportion. We use it to estimate a probability; it may differ from the theoretical probability of a model.
Do I include earlier trials when predicting further results?
Use earlier trials to estimate the probability, then multiply by the stated number of future trials. Include earlier counts only if the question asks for an overall total.