Basic Probability
Probability
Relative frequency
- Some probabilities cannot be worked out in advance, so they are estimated from an experiment instead
- Relative frequency = number of successes ÷ total number of trials
- It is only an estimate, and the more trials are run the closer it should get to the true probability
- Comparing a relative frequency with the theoretical probability is how a spinner or dice is judged fair or biased
- A fair six-sided dice should give a five about 1/6 of the time, roughly 0.167
- Rolling 300 times and getting 48 fives gives a relative frequency of 48 ÷ 300 = 0.16, close enough to suggest the dice is fair
- A relative frequency far from the theoretical value suggests bias
Expected frequency
- Going the other way, a known probability predicts how often something should happen in a given number of trials
- Expected frequency = probability × number of trials
- The answer is a prediction rather than a guarantee, so the actual number will usually differ a little
- A non-whole answer is acceptable, since an expected frequency is an average rather than a count
- With a probability of 0.15 over 400 trials, the expected frequency is 0.15 × 400 = 60
Exam tip
Use your own probability for the expected number
When a question asks for a probability and then for the expected number of times in so many trials, answer the second part even if you doubt the first. The schemes give that mark for your own probability multiplied by the number of trials. So multiply exactly what you wrote, but check it first: some schemes follow it through only when it lies between 0 and 1, so a value above 1 means a slip to find.
Worked example
Relative frequency and expected frequency together
A spinner has five equal sections numbered 1 to 5. In an experiment it is spun 200 times and lands on 4 on 32 occasions. Find the relative frequency of landing on 4, say whether the spinner appears to be fair, and find the expected number of 4s in 500 spins of a fair spinner.
Solution:
- Relative frequency = 32 ÷ 200 = 0.16
- For a fair five-section spinner the theoretical probability is 1 ÷ 5 = 0.2
- 0.16 is reasonably close to 0.2, so on this evidence the spinner appears roughly fair, though more trials would give a better test
- For a fair spinner, expected frequency = probability × number of trials
- Expected number of 4s = 0.2 × 500 = 100
- Had the question asked how many 4s to expect in 500 spins of this spinner, rather than a fair one, the relative frequency would be the probability to use: 0.16 × 500 = 80