Relative & Expected Frequency (Edexcel GCSE Maths: Foundation)

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Relative Frequency

What is relative frequency?

  • Relative frequency is an estimate of a probability using results from an experiment
    • For a certain number of trials of that experience, the probability of ‘success’ is:

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

    • If you flip an unfair coin 50 times and it lands on heads 20 times, an estimate for the probability of the coin landing on heads is  20 over 50 (its relative frequency)
      • That is the best estimate we can make, given the data we have
      • We do not know the actual probability
  • The more trials that are carried out, the more accurate relative frequency becomes
    • It gets closer and closer to the actual probability 

When will I be asked to use relative frequency?

  • Relative frequency is used when actual probabilities are unavailable (or not possible to calculate)
    • For example, if you do not know the actual probability of being left-handed, you can run an experiment to find an estimate (the relative frequency)
  •  Sometimes actual probabilities are known, as they can be calculated in theory (called theoretical probabilities)
    • The theoretical probability of a fair coin landing on heads is 0.5
    • The theoretical probability of a fair standard six-sided dice landing on a six is 1 over 6 
  • Relative frequency can be compared to a theoretical probability to test if a situation is fair or biased 
    • If 100 flips of the coin give a relative frequency of 0.48 for landing on heads, the coin is likely to be fair
      • The theoretical probability is 0.5 and 0.48 is close to 0.5
    • If 100 flips of the coin give a relative frequency of 0.13 for landing on heads, the coin is likely to be biased (not fair)

What else do I need to know about relative frequency?

  • Relative frequency assumes that there is an equal chance of success on each trial
    • The trials are independent of each other
      • For example, if choosing something out of a bag (a ball, or marble etc), it would need to be replaced each time to use relative frequency

Examiner Tip

  • Exam questions will not necessarily use the phrase relative frequency.
  • If you have to choose the best estimate, choose the one with the most trials.

Worked example

There are an unknown number of different coloured 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 you would expect a particular outcome to occur 
  • It is found by multiplying the probability by the number of trials
    • If you flip a fair coin 100 times, you would expect 0.5 × 100 = 50 heads
  • Sometimes you need to calculate the relative frequency first
    • If you flip a biased coin 40 times and get 10 heads, how many heads would you expect when flipping 100 times?
      • The relative frequency is 10 over 40 = 0.25 from the first experiment
      • 0.25 × 100 = 25, you would expect to get heads 25 times from 100 throws

Examiner Tip

  • Exam questions will not necessarily use the phrase "expected frequency", but might ask how many you "would expect".

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 this experiment is repeated 300 times?

 

This is expected frequency so multiply the number of trials by the probability from part (a)

300 cross times 1 third equals 100

We would expect 100 yellow counters

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Mark

Author: Mark

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.