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Cambridge IGCSE0607

Relative and expected frequencies

International Mathematics 0607 Chapter Notes

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Relative and expected frequencies
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In this chapter

  1. 1Relative frequency
  2. 2More trials give a better estimate
  3. 3Expected frequency
  4. 4Finding the probability from a ratio
  5. 5Probabilities in a table
  6. 6Fair and biased

1. Relative frequency

In an experiment, the frequency of an outcome is how many times it happens. The relative frequency is that frequency divided by the total number of trials.

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

The relative frequencies of all the outcomes add up to 1.

relative frequency = (frequency of the outcome)/(total number of trials)

Examiner insight

“Some candidates confused the information with cumulative frequency and give incorrect answers of 60, 110, 156, 200.” “Many did not understand the term ‘relative frequency’. Consequently, many divided the frequencies by 100 instead of dividing by 200.”

Worked example 11 mark

A 4-sided spinner is spun 200 times. The results are given in the table. Complete the table to show the relative frequencies.

number on spinner1234
frequency60504644
relative frequency
  1. 1

    Divide each frequency by the total, 200: 60/200, 50/200, 46/200, 44/200 (or equivalent) [1]

2. More trials give a better estimate

Relative frequency estimates a probability. The estimate is better when the experiment has a large number of trials.

A short run of trials can give a relative frequency that is far from the true probability. A long run settles close to it.

To justify an estimate, say that there were a sufficient number of trials. Do not say only that the relative frequency stayed the same.

Examiner insight

“Many candidates recognised that a good estimate comes from completing a large or sufficient number of trials and were awarded the mark.” “Candidates should know that accuracy is improved every time you carry out an experiment.”

Worked example 12 marks

A bag holds red and blue beads. Sam picks a bead, records its colour and replaces it. In his first 20 picks he gets 5 red. In 400 picks he gets 140 red. Use the 400 picks to estimate the probability of red. Give a reason why this is a better estimate than the first 20 picks.

  1. 1

    Estimate = 140/400 = 0.35 [1]

  2. 2

    It is based on a large number of trials [1]

3. Expected frequency

An expected frequency is how many times an outcome should happen in a number of trials.

Multiply the probability of the outcome by the number of trials. The result is a number of times, not a probability.

The probability that an outcome does not happen is 1 minus the probability that it does.

expected frequency = probability × number of trials

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

Examiner insight

“Candidates needed to multiply their probability from part (a) by 140.” “If these candidates had thought about this answer they should have realised that it could not be correct.”

Worked example 12 marks

The probability that any train arrives at station A on time is 5/6.(a) Write down the probability that a train arriving at station A is not on time.(b) One day 72 trains arrive at station A. Work out the number of these trains that are expected to arrive on time.

  1. 1

    (a) 1 - 5/6 = 1/6 [1]

  2. 2

    (b) 72 × 5/6 = 60 [1]

4. Finding the probability from a ratio

Sometimes the probability is hidden in a ratio. Add the parts of the ratio to find the total. The probability of an outcome is its part over the total. Multiply by the number of trials for the expected frequency.

The total number of items must be a multiple of the sum of the parts.

ratio a : b ⇒ P(b) = b/(a + b)

Examiner insight

“To find the expected number of defective rivets, candidates had to multiply their probability from part (a) by 10 000.”

Worked example 13 marks

A bag contains red counters and green counters. The ratio red counters : green counters = 2 : 3.(a) Sammy takes a counter at random from the bag and replaces it. Sammy repeats this 200 times. Work out the expected number of times Sammy takes a green counter.(b) Maddy says that the total number of counters in the bag is 88. Explain why Maddy must be wrong.

  1. 1

    (a) Total parts = 2 + 3 = 5, so P(green) = 3/5 [1]

  2. 2

    3/5 × 200 = 120 [1]

  3. 3

    (b) The total must be a multiple of 5, and 88 is not a multiple of 5 [1]

5. Probabilities in a table

The probabilities of all the outcomes add up to 1. The same is true of relative frequencies.

To find a missing probability, subtract the others from 1.

When a question says estimate, use the probabilities. A guess such as "about 400" earns nothing.

missing probability = 1 - (sum of the other probabilities)

Examiner insight

“Many candidates did not seem to understand that the relative frequencies sum to 1.” “A final point, when a question asks for candidates ‘to work out an estimate’, then a guess or answer like ‘about 400’ is not correct as this instruction means ‘use probabilities to work out’.”

Worked example 12 marks

Sundeep throws a biased 6-sided die. The table shows the probability of the die landing on each number.(a) Find the value of x.(b) Sundeep throws this die 180 times. Find the expected number of times this die lands on 4.

number123456
probability11/7213/728/7210/7214/72x/72
  1. 1

    (a) 11 + 13 + 8 + 10 + 14 = 56, so x = 72 - 56 = 16 [1]

  2. 2

    (b) 180 × 10/72 = 25 [1]

6. Fair and biased

A fair die, coin or spinner gives each outcome the same probability. Otherwise it is biased.

For a fair 6-sided die each face has probability 1/6. Compare the relative frequencies from an experiment with the expected values.

A big difference over a large number of trials is evidence of bias. A big difference over a few trials is not.

If a die is biased, do not go back to using 1/6. Use the probability the question gives.

Examiner insight

“Most candidates realised that the die was biased because of the wide variation in the frequencies.” “However, few mentioned that this is only significant because this is in a large number of trials.”

Worked example 13 marks

A 6-sided die is rolled 600 times. It lands on 6 a total of 150 times. Is there evidence that the die is biased? Give a reason.

  1. 1

    If the die is fair, expected number of 6s = 600 × 1/6 = 100 [1]

  2. 2

    The result, 150, is much higher than 100 [1]

  3. 3

    This is from a large number of trials, so the die is probably biased [1]

Recap

  • Relative frequency = frequency ÷ total number of trials. It estimates probability.
  • More trials give a better estimate.
  • Expected frequency = probability × number of trials.
  • Probabilities of all outcomes add up to 1, so P(not A) = 1 − P(A).
  • From a ratio a : b, P(b) = b ÷ (a + b).
  • A fair object has equal probabilities. A big gap over many trials suggests bias.

End-of-chapter exercise

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

  1. Write down how to work out the relative frequency of an outcome in an experiment.1 mark
  2. Do not use a calculator. A coin is flipped 50 times and lands on heads 28 times. Write the relative frequency of heads as a fraction in its simplest form.2 marks
  3. Do not use a calculator. A spinner is spun 80 times. The results are given in the table. Write each relative frequency as a decimal. | number on spinner | 1 | 2 | 3 | 4 | | --- | --- | --- | --- | --- | | frequency | 10 | 30 | 24 | 16 |2 marks
  4. Do not use a calculator. The probability that a bus is late is 0.2. There are 150 buses. Work out the number of buses expected to be late.2 marks
  5. Do not use a calculator. A bag contains only red counters and blue counters in the ratio 3 : 5. A counter is taken at random and replaced. This is done 400 times. Work out the expected number of times a blue counter is taken.3 marks
  6. A biased coin is flipped 240 times and lands on tails 132 times. (a) Find the relative frequency of tails as a decimal. (b) Estimate the number of tails in 1000 flips.3 marks
  7. Do not use a calculator. A biased 5-sided spinner lands on 1, 2, 3, 4 or 5. The probabilities of 1, 2, 3 and 4 are 0.1, 0.25, 0.3 and 0.15. The spinner is spun 400 times. Work out the expected number of times it lands on 5.3 marks
  8. A bag holds 60 counters, each red or yellow. A counter is taken at random and replaced. This is done 300 times and red is taken 105 times. (a) Estimate the probability of taking a red counter. (b) Estimate the number of red counters in the bag. (c) Estimate the number of times a yellow counter is taken in 1000 further trials.5 marks
  9. A games company tests a 6-sided die by rolling it 480 times. The results are given in the table. (a) Work out the number of times each number is expected if the die is fair. (b) Do the results suggest the die is biased? Give a reason. (c) The same die is rolled 1500 more times. Estimate the number of times it lands on 6. | number | 1 | 2 | 3 | 4 | 5 | 6 | | --- | --- | --- | --- | --- | --- | --- | | frequency | 62 | 70 | 58 | 66 | 64 | 160 |5 marks
  10. A machine gives one toy at random each time it is used: a car, a robot, a ball or a ring. Priya records the toys from 250 uses. The table gives some of her relative frequencies. (a) Work out the number of times the machine gave a ball. (b) The owner wants the machine to give out 240 rings. Estimate the number of times the machine must be used. (c) The owner says each toy is equally likely. Does Priya's data support this? Give a reason. | toy | car | robot | ball | ring | | --- | --- | --- | --- | --- | | relative frequency | 0.34 | 0.18 | | 0.16 |7 marks

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