Relative frequency and expected frequency
Introduced| English | Meaning |
|---|---|
| bias/ˈbaɪəs/ | A systematic tendency for a chance device to favour particular outcomes rather than behave as intended. |
| relative frequency/ˈrelətɪv ˈfriːkwənsi/ | The number of occurrences of an outcome divided by the total number of observations or trials. |
| expected frequency/ekˈspektɪd ˈfriːkwənsi/ | The predicted count of an outcome: probability × number of trials. |
Is the coin fair?
- You flip a coin 100 times and get 60 heads. Is the coin biased 偏倚?
- Theoretical probability says $\dfrac{1}{2}$. Experimental probability (the relative frequency 相对频率) says $\dfrac{60}{100} = 0.6$. More flips usually give a more stable estimate, but no finite experiment proves the probability with certainty.
Relative frequency
- When outcomes are not equally likely, run an experiment:
- More trials → a better estimate of the true probability.
- A fair object gives equal chances; one with bias does not.
A drawing pin is dropped 200 times and lands point-up $84$ times. Relative frequency of point-up $= \dfrac{84}{200} = 0.42$.
A sample space diagram lists every outcome; for two dice there are $36$ equally likely totals
Two-dice probability
Roll the two dice many times: the bars start jumpy but settle into the theoretical triangle peaking at 7 — experimental probability closing in on theory.
Relative frequency is calculated as:
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Relative frequency = successes ÷ number of trials.
Relative frequency becomes a better estimate as the number of trials increases.
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More trials → the relative frequency converges to the true probability.
A drawing pin is dropped 200 times and lands point-up 84 times. The relative frequency of point-up is (to 2 dp):
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84/200 = 0.42.
Expected frequency 期望频数
- How many times you expect an event in $n$ trials:
Expected ≠ guaranteed. An expected frequency of $50$ sixes in $300$ rolls doesn't mean you'll get exactly $50$. It's the average over many sets of $300$ rolls.
The probability of rolling a six is 1/6. How many sixes are expected in 300 rolls?
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(1/6) × 300 = 50.
An event has probability 0.2. How many times is it expected in 50 trials?
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0.2 × 50 = 10.
Expected frequency = P(event) × number of ______.
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Expected frequency = probability × number of trials.
Worked example
- $\text{P}(\text{six}) = \dfrac{1}{6}$. In $300$ rolls: expected $= \dfrac{1}{6} \times 300 = 50$ sixes.
- $\text{P}(\text{rain}) = 0.2$. In $50$ days: expected $= 0.2 \times 50 = 10$ rainy days.
Expected frequency grows linearly with trials: double the trials, double the expected count.
You've got it
- relative frequency $= \dfrac{\text{times happened}}{\text{total trials}}$ — improves with more trials
- expected frequency $= \text{P}(\text{event}) \times n$
- $\dfrac{1}{6} \times 300 = 50$ expected sixes
Write from an expected number of 20
- The probability of success is 1/4.
- There are 80 trials.
The probability of an event is 1/4. In 80 trials, how many times is it expected?
A quarter of 80 is 20.
What is the expected number?
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Probability times trials is 20. Other options — 50 (the lesson expected count); 80 (the number of trials).