Relative & Expected Frequency (CIE IGCSE Maths: Extended)

What is relative frequency?

• Relative frequency is used to estimate probabilities from experimental data
  • For a certain number of trials, the probability of ‘success’ is given as

• • e.g. If you flip an unfair coin 50 times and it lands on heads 20 times then you would use relative frequency to estimate the probability of the coin landing on heads as 
• The more trials that are carried out, the more accurate relative frequency becomes
  • If you have to choose between relative frequencies to estimate the probability then choose the one which includes the largest number of trials

When will I be asked to use relative frequency?

• Relative frequency will be used when either theoretical probabilities are unavailable or are not possible to calculate
• Relative frequency may also be used to test if a situation is fair or biased
  • e.g.  if a coin is fair then you would expect the probability of it landing on heads to be 0.5
    • If the relative frequency is close to 0.5 then this suggests it is fair
    • If the relative frequency is not close to 0.5 then this suggests the coin is biased (not fair)

What else do I need to know about relative frequency?

• Relative frequency will only provide an estimate for a probability
  • If you use a large number of trials then you would expect the estimate to be close to the actual probability
• Relative frequency assumes that there is an equal chance of ‘success’ on each trial
  • i.e. trials are independent
  • if choosing something from a bag (button, ball, marble) then it would need to be replaced to use relative frequency

What is expected frequency?

• Expected frequency refers to the number of times a you would expect a particular outcome to occur when repeating a trial numerous times
• The theoretical probability of that outcome will need to be known
• or an estimate of it, from relative frequency

How do I find expected frequency?

• If the probability of a particular outcome is p and there are n trials then
• the expected number of occurrences of that outcome from the n trials is np
• multiply the number of trials by the probability of the particular outcome
• Note that this does not mean that there will exactly np occurrences
• but if the experiment (of n trials) was repeated over and over again we would expect the number of occurrences to average out to be np