Mathematics (Core)Exam code: 0580
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
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