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 perfectly acceptable and should not be rounded unless the question asks
    • With a probability of 0.15 over 400 trials, the expected frequency is 0.15 × 400 = 60
Exam tip

Write the fraction unsimplified first, so the count is visible, then simplify. Fractions, decimals and percentages are all accepted — but never a ratio, and never "4 out of 36" on its own. Relative frequency is an estimate from an experiment; expected frequency is a prediction for future trials, found by multiplying probability by the number of trials, and the two are easy to confuse.

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