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Relative & Expected Frequency (Cambridge O Level Maths)

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Dan

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Dan

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Relative Frequency

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

fraction numerator Number space of space successful space outcomes over denominator Total space number space of space trials end fraction

    • 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 20 over 50
  • 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

Examiner Tip

  • Exam questions will not necessarily use the phrase relative frequency so think about the information given carefully
  • If a question mentions repeatedly carrying out a trial, or experiment, or the possibility of bias, then relative frequency is involved

Worked example

There are an unknown number of different coloured but identically sized buttons in a bag.  Johan selects a button at random, notes its colour and replaces the button in the bag.  Repeating this 30 times, Johan notes that on 18 occasions he selected a red button.

Use Johan’s results to estimate the probability that a button drawn at random from the bag is red.

Taking ‘red’ to be a success, Johan had 18 successes out of a total of 30 trials.

bold P stretchy left parenthesis red stretchy right parenthesis bold equals bold 18 over bold 30 bold equals bold 3 over bold 5

Expected 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

Examiner Tip

  • Exam questions will not necessarily use the phrase expected frequency so think about the information given carefully
  • If a question mentions repeatedly carrying out a trial, and a number of occurrences is requested (rather than a probability) expected frequency is involved

Worked example

There are 6 blue, 4 red and 5 yellow counters in a bag.  One counter is drawn at random and its colour noted.  The counter is then returned to the bag.

a)

Find the probability that a counter drawn from the bag is yellow.

There are 5 yellow counters out of a total of 6 + 4 + 5 = 15 counters in the bag.

P(Yellow)bold equals bold 5 over bold 15 bold equals bold 1 over bold 3

b)

How many times would you expect a yellow counter to be drawn if the experiment is repeated 300 times?

This is expected frequency so multiply the number of trails (n) by the probability (p).

300 cross times 1 third equals 100

We would expect 100 yellow counters

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    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.