Estimating Probabilities
Learn to estimate probabilities from data for Queensland Year 11 Mathematical Methods (QCAA). When outcomes are not equally likely, the relative frequency of an event over many trials estimates its probability.
You will learn to read frequency tables, treat relative frequencies as point estimates of probabilities, compute an expected number as probability times trials, and watch estimates settle as trials increase — groundwork for conditional probability and independence.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA, Unit 1 Topic 5), an experimental probability is estimated from data using the relative frequency \(\dfrac{\text{frequency}}{\text{total trials}}\). This page shows how to estimate a probability from a table or bar chart, predict the expected number of occurrences, and compare experimental with theoretical probability.
The relative frequency of an outcome is how often it happened divided by the number of trials: \(\dfrac{\text{frequency}}{\text{total trials}}\). We use it as an experimental (estimated) probability when the true probability is unknown or the outcomes are not equally likely.
The expected number of times an outcome occurs in \(n\) trials is the estimated probability multiplied by \(n\). It is a prediction of a count, so we usually round to a whole number.
By the law of large numbers (used informally here), the relative frequency settles closer to the true probability as the number of trials grows — a larger sample gives a more reliable estimate.
| Colour | Red | Blue | Green | Total |
|---|---|---|---|---|
| Frequency | 45 | 30 | 25 | 100 |
The experimental probability (relative frequency):
The expected number of occurrences in \(n\) trials:
How to estimate and use a probability from data
- Read the frequency of the outcome and the total number of trials.
- Divide: experimental probability \(=\dfrac{\text{frequency}}{\text{total}}\).
- Predict a count: expected number \(=\) probability \(\times\) number of trials.
- Compare or scale: line the estimate up against the theoretical value, or use the complement \(P(\text{not})=1-P\) to answer the question.
Relative frequency \(=\dfrac{\text{frequency}}{\text{total}}\):
| \(P(\text{up})\) | \(=\) | \(\dfrac{130}{200}\) |
Simplify to a decimal:
| \(=\) | \(\dfrac{13}{20}\) | |
| \(=\) | \(0.65\) |
\(P(\text{up})\approx 0.65\).
Expected number \(=P\times n\):
| \(\text{expected}\) | \(=\) | \(0.04\times 1500\) |
Multiply:
| \(=\) | \(60\) |
About \(60\) faulty components are expected.
| Outcome | Faulty | Not faulty |
|---|---|---|
| Estimated probability | \(0.04\) | \(0.96\) |
Experimental probability from the data:
| \(P_{\text{exp}}\) | \(=\) | \(\dfrac{14}{60}\) |
| \(=\) | \(\dfrac{7}{30}\approx 0.23\) |
Theoretical probability for a fair die:
| \(P_{\text{theory}}\) | \(=\) | \(\dfrac{1}{6}\approx 0.17\) |
The experimental value \(0.23\) is a little higher than the theoretical \(0.17\); with more rolls it would be expected to settle closer to \(\dfrac{1}{6}\).
\(P_{\text{exp}}=\dfrac{7}{30}\approx 0.23\), above \(P_{\text{theory}}=\dfrac{1}{6}\approx 0.17\).
| Score on the die | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Frequency | 8 | 11 | 9 | 10 | 14 | 8 |
Estimate \(P(\text{red})\) from the table:
| \(P(\text{red})\) | \(=\) | \(\dfrac{45}{100}\) |
| \(=\) | \(0.45\) |
Complement rule for ‘not red’:
| \(P(\text{not red})\) | \(=\) | \(1-0.45\) |
| \(=\) | \(0.55\) |
Scale up to \(400\) spins:
| \(\text{expected reds}\) | \(=\) | \(0.45\times 400\) |
| \(=\) | \(180\) |
\(P(\text{red})=0.45\), \(P(\text{not red})=0.55\), and about \(180\) reds in \(400\) spins.
| Colour | Red | Blue | Green | Total |
|---|---|---|---|---|
| Frequency | 45 | 30 | 25 | 100 |
Common pitfalls
Frequently asked questions
What is relative frequency?
How often an outcome occurred divided by the number of trials: \(\dfrac{\text{frequency}}{\text{total}}\). It is used as an experimental probability.
What is the difference between experimental and theoretical probability?
Experimental probability is estimated from data (relative frequency); theoretical probability is calculated from equally likely outcomes.
How do you find the expected number of occurrences?
Multiply the estimated probability by the number of trials: expected number \(=P\times n\).
Why does more data give a better estimate?
By the law of large numbers the relative frequency settles closer to the true probability as the number of trials increases.
Can experimental probability equal theoretical probability?
It can, but usually they differ slightly. They tend to agree more closely as the number of trials grows.
Do relative frequencies have to add to 1?
Yes — across all outcomes the relative frequencies (like all probabilities) add to \(1\).