Relative frequency and expected frequency

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 sample space diagram lists every outcome; for two dice there are $36$ equally likely totals

Expected frequency 期望频数

• How many times you expect an event in $n$ 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

Write from an expected number of 20

• The probability of success is 1/4.
• There are 80 trials.