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Cambridge IGCSE0580

Relative and expected frequencies

Mathematics 0580 Chapter Notes

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Relative and expected frequencies
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In this chapter

  1. 1Relative frequency estimates probability
  2. 2Expected frequency
  3. 3Fair and biased
  4. 4Find the probability first

1. Relative frequency estimates probability

Do an experiment, such as spinning a spinner. Relative frequency is the number of times a result happens divided by the total number of trials.

Relative frequency is an estimate of the probability of that result. It can be written as a fraction, a decimal or a percentage.

The relative frequencies of all the possible results add up to 1.

relative frequency = (number of times the result happens)/(total number of trials)

Examiner insight

“Some candidates did not fully understand the term relative frequency. This could be expressed as a fraction, decimal or percentage as the question did not specify the form of the answer.” “A significant proportion of candidates showed no understanding of probability and gave frequencies rather than relative frequencies as their answers.”

Worked example 11 mark

Reem has a six-sided spinner numbered 1 to 6. Reem spins the spinner 50 times. Her results are in the table.

(a) Write down the relative frequency of the spinner landing on the number 4.

number on spinner123456
frequency8759129
  1. 1

    relative frequency = (frequency of 4)/(total spins) = 9/50 [1]

2. Expected frequency

A probability tells you what to expect in a larger number of trials. The expected number is the probability multiplied by the number of trials.

The probability can come from a given value, or be estimated from a relative frequency.

The answer is a number of times, so it is not a probability. Do not give it as a fraction out of the number of trials.

expected frequency = probability × number of trials

Examiner insight

“Multiplying 3 by 160 to produce 480 suggests those candidates did not appreciate that there were only 160 spins.” “However, some did lose a mark by converting the probability to a decimal without sufficient figures for accuracy when multiplied by 570.”

Worked example 12 marks

Rishwan writes each of these numbers on a card.

-50 -11 9 15 27 50 83

Rishwan picks a card at random and replaces it.

(i) Find the probability that he picks a natural number.

(ii) He repeats this 140 times. Find the expected number of times that Rishwan picks a natural number.

  1. 1

    natural numbers: 9, 15, 27, 50, 83, so 5 of the 7 cards: probability = 5/7 [1]

  2. 2

    expected number = 5/7 × 140 = 5 × 20 = 100 [1]

3. Fair and biased

A fair spinner or dice has all its outcomes equally likely. For a fair six-sided dice each number has probability 1/6.

A biased one does not: some outcomes happen more often than others.

To test it, find the expected frequency for a fair one and compare it with the results. A result far from the expected number suggests bias.

Worked example 13 marks

Do not use a calculator.

A dice is rolled 60 times. A six comes up 20 times. Is there evidence that the dice is biased? Give a reason.

  1. 1

    fair dice: expected sixes = 1/6 × 60 = 10 [1]

  2. 2

    relative frequency of a six = 20/60 = 1/3, twice 1/6 [1]

  3. 3

    20 is far more than 10, so the dice may be biased towards six [1]

4. Find the probability first

Sometimes the probability must be found before you can use it. The probabilities of all the outcomes add up to 1.

So the probability that something does not happen is 1 minus the probability that it does. The same sum finds a missing probability in a table.

Read which outcome the question asks for, then multiply.

P(not A) = 1 - P(A)

Examiner insight

“Subtracting the decimal from 1 was often given as 99.3 or (100 – 0.3) or left as 100 – 0.7.” “Those who knew the probabilities added to 1 generally got the correct answer.”

Worked example 14 marks

The probability that a team wins a match is 0.4.

(a) The team plays 20 matches. Work out the expected number of matches they win.

(b) The probability that the team loses is three times the probability that it draws. Complete the table.

resultwinlosedraw
probability0.4
  1. 1

    (a) 0.4 × 20 = 8 [1]

  2. 2

    (b) lose and draw together: 1 - 0.4 = 0.6 [1]

  3. 3

    lose is 3 parts, draw is 1 part: 4 parts, so draw = 0.6 ÷ 4 = 0.15 [1]

  4. 4

    lose = 3 × 0.15 = 0.45 [1]

Recap

  • Relative frequency = number of times a result happens ÷ total number of trials.
  • Relative frequency is an estimate of probability. It can be a fraction, decimal or percentage.
  • Expected frequency = probability × number of trials.
  • A fair dice or spinner has equally likely outcomes. A biased one does not.
  • Probabilities of all outcomes add up to 1, so P(not A) = 1 − P(A).

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. A coin is thrown 40 times. It lands on heads 22 times. Write down the relative frequency of heads as a fraction in its simplest form.2 marks
  2. Do not use a calculator. A spinner is spun 80 times. It lands on red 28 times and on blue 36 times. The only other colour is green. Find the relative frequency of green.3 marks
  3. The probability that a seed grows into a plant is 0.8. A gardener plants 350 seeds. How many seeds are expected not to grow?3 marks
  4. Do not use a calculator. A counter is taken from a bag and replaced. This is done 120 times. A blue counter is taken 30 times. (a) Estimate the probability of taking a blue counter. (b) The counter is taken and replaced 500 times. Estimate the number of times a blue counter is taken.3 marks
  5. Do not use a calculator. A fair six-sided dice is rolled 90 times. Work out the number of times it is expected to show a number greater than 4.3 marks
  6. A factory tests 250 bulbs. 7 are faulty. Estimate the number of faulty bulbs in a batch of 12000.3 marks
  7. Do not use a calculator. A coin is thrown 200 times. It lands on heads 130 times. Work out the relative frequency of heads. Does this suggest the coin is biased? Give a reason.3 marks
  8. Do not use a calculator. In a survey, 28 out of 80 students say they walk to school. The school has 1200 students. Estimate how many of the 1200 students do not walk to school.4 marks
  9. Do not use a calculator. A machine sorts apples into four groups: small, medium, large and rejected. The probability that an apple is rejected is 0.08. The probability that it is small is 0.2. The probability that it is medium is three times the probability that it is large. The machine sorts 4500 apples. Work out the expected number of medium apples.4 marks
  10. A dice is rolled 150 times. The relative frequency of a six is 0.22. The dice is rolled 250 more times. For all 400 rolls, the relative frequency of a six is 0.19. (a) Work out the number of sixes in the last 250 rolls. (b) Use the results of all 400 rolls to estimate the number of sixes in 3000 rolls. (c) Work out how many more sixes this is than the number expected from a fair dice in 3000 rolls.5 marks

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