Experimental probability, fairness and expected outcomes · Higher
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| relative frequency/ˈrelətɪv ˈfriːkwənsi/ | 相对频率 | xiāng duì pín lǜ |
A coin gives six heads in its first ten tosses. This does not mean it must give four tails next, or prove that the coin is unfair.
- A coin gives six heads in its first ten tosses. This does not mean it must give four tails next, or prove that the coin is unfair.
- This lesson studies relative frequency 相对频率: The observed count of an outcome divided by the number of trials.
Choose the mathematical structure
- Record each trial consistently and total the counts. Relative frequency estimates probability as outcome frequency/trials. For a stated probability p, expected count in n future trials is np; it is a long-run average, not a guarantee. Unbiased trials with larger samples usually give more stable estimates, but cannot force exact agreement with theory. Probabilities lie from 0 to 1 and a mutually exclusive exhaustive list sums to 1.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines relative frequency?
The observed count of an outcome divided by the number of trials.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
A spinner lands red 18 times in 60 spins, giving estimated P(red)=18/60=0.3=30%. Using this estimate predicts 0.3×200=60 red results in 200 future spins. If red, blue and green are exhaustive with probabilities 0.3,0.45 and p, then p=1-0.75=0.25. A fair coin has theoretical P(head)=0.5; 100 tosses give expected heads 50, but 48 or 54 is possible. An experiment with 6 heads in 10 tosses estimates 0.6; one with 502 heads in 1000 estimates 0.502. These illustrative runs are not proof that error falls at every stage. Spin using the same method and record all results rather than stopping when a favourite outcome appears.
Experimental probability, fairness and expected outcomes
Record each trial consistently and total the counts
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find red relative frequency for 18 red results in 60 trials.
18/60=0.3.
Test a tempting shortcut
- A larger biased sample can still be misleading. Expected does not mean certain. Mutually exclusive events cannot happen together; if categories overlap, do not simply add their probabilities as separate outcomes.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Expected count 50 means exactly 50 successes must occur. This claim is false. Explain which definition or assumption it violates.
Use p=0.3 to estimate red count in 200 trials.
Expected count=np=200×0.3=60.
Expected count 50 means exactly 50 successes must occur.
A larger biased sample can still be misleading. Expected does not mean certain. Mutually exclusive events cannot happen together; if categories overlap, do not simply add their probabilities as separate outcomes.
Interpret a new situation
- AQA P1–P5 uses frequency tables/trees, randomness/fairness, expected counts, the probability scale and empirical/theoretical comparison. State whether a value is observed, estimated or theoretical.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Red,blue,green probabilities are 0.3,0.45,p. Find p.
Exhaustive separate outcomes sum to 1: p=1-0.3-0.45.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.5. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The observed count of an outcome divided by the number of trials. Choose the relationship, show the method, check its assumptions and interpret the result.