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Relative Frequency and Expected Outcomes - Edexcel GCSE Higher Maths

Typically 2-5 marks per paper, usually one question set in two or three parts · Spec 1MA1

Relative frequency is the share of trials in which something actually happened. Roll a dice 200 times, count 46 sixes, and 46200=0.23\frac{46}{200} = 0.23 is your estimate for the probability of a six. Edexcel 1MA1 files this under three spec points that are easiest to read as one idea: P2, calculating expected outcomes of future experiments from ideas of randomness and fairness; P3, relating relative expected frequencies to theoretical probability on the 0 to 1 scale; and P5, that samples from an unbiased experiment tend towards the theoretical distribution as the sample size increases. All three are common content, so a Foundation candidate meets the same material.

The set-up is almost always a record of results: a list of outcomes with their frequencies, or a running count after 10, 50, 100 and 500 trials. Part (a) wants an estimate of one probability, 1-2 marks, and is a single division. Part (b) scales that estimate to a new number of trials, 2 marks, and is a single multiplication. Part (c) is the judgement, and it carries the give-away instruction 'give a reason for your answer': is the spinner biased, or which of two students has the more reliable estimate. It can appear on any of the three papers. The calculator makes little difference here, though on Paper 1 you are more likely to be left holding a fraction such as 1930\frac{19}{30}, which Edexcel accepts unsimplified.

The reason mark is the one that disappears. 'It landed on 4 more than the others' describes the table rather than testing it, and scores nothing; the examiner is looking for the observed figure set beside what a fair object would produce, such as 82 fours against an expected 50 in 200 spins. Two other losses recur. When results from two experiments are combined, add the successes and add the trials; averaging the two relative frequencies gives a different number and no marks. And when a question asks which estimate is best, the answer is the one from the largest number of trials, not the one with the biggest frequency or the tidiest decimal.

Worked example

A five-sided spinner has equal sections numbered 1, 2, 3, 4 and 5. Nadia thinks the spinner is biased towards 5. Nadia spins the spinner 40 times. It lands on 5 fourteen times. Rob spins the same spinner 160 times. It lands on 5 forty-six times. (a) Using all of the results, work out the best estimate for the probability that the spinner lands on 5. (2) (b) The spinner is to be spun 500 more times. Work out an estimate for the number of times it will land on 5. (2) (c) Do the results support Nadia's claim? Use figures to justify your answer. (1)
[5 marks]
  1. Pool the two experiments: 14+46=6014 + 46 = 60 fives out of 40+160=20040 + 160 = 200 spins.
    M1: adding the successes and adding the trials, not averaging the two relative frequencies
  2. The best estimate is 60200=0.3\frac{60}{200} = 0.3.
    A1: 0.3 or 3/10 or 60/200; unsimplified fractions are accepted
  3. For 500 further spins, multiply the estimate by the number of trials: 0.3×5000.3 \times 500.
    M1: their probability times 500, follow through from part (a)
  4. 0.3×500=1500.3 \times 500 = 150 times.
    A1: 150; an answer of 0.3 here scores the method mark only
  5. A fair spinner gives P(5)=15=0.2P(5) = \frac{1}{5} = 0.2, so 200 spins would produce about 40 fives. Nadia and Rob recorded 60, half as many again, over a large number of trials, so the results do support bias towards 5.
    B1: comparison of 60 with the expected 40 (or 0.3 with 0.2); a comment with no figures scores nothing

Practice questions

These are original questions in Edexcel 1MA1 Higher style. Nothing is printed as a table or a picture, so each set of results is written out in a line or two: read for the number of trials, which is your denominator, and for the frequency of the outcome you are asked about. Type probabilities as fractions if that is what your working gives (19/30 marks correct), and where the exact value does not terminate the stem asks for 3 decimal places instead. The written 'give a reason' parts appear here as multiple choice, and the wrong options are the ones that reach the right verdict by the wrong route.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Exam pace [2 marks]
    Amir rolls a biased dice and keeps a running count of the number of 4s. After 20 rolls he has recorded 7 fours. After 100 rolls he has recorded 27 fours. After 400 rolls he has recorded 96 fours. Write down the best estimate for the probability that the dice lands on 4.
  2. Question 2 Exam pace [3 marks]
    A four-sided spinner has equal sections labelled A, B, C and D. Priya spins it 200 times. It lands on A 38 times, on B 41 times, on C 39 times and on D 82 times. Priya says the spinner is biased. Which reason best supports what she says?
    Answer options for question 2
  3. Question 3 Exam pace [3 marks]
    In a survey of 240 vehicles passing a school, 42 were vans. The next 1000 vehicles to pass the school are recorded. Work out an estimate for the number of these that will be vans.
  4. Question 4 Exam pace [2 marks]
    A machine fills bottles with juice. In a sample of 150 bottles, 12 contained less than 500 ml. Work out an estimate for the probability that a bottle from this machine contains less than 500 ml.
  5. Question 5 Stretch [4 marks]
    A game uses a square target of side 20 cm. A circle of radius 8 cm is drawn inside the square. A player throws 250 darts, and every dart lands at a random position somewhere on the square. Work out an estimate for the number of darts that land inside the circle. Give your answer to the nearest whole number.
  6. Question 6 Stretch [4 marks]
    A spinner can land on red, blue or green only. The spinner is spun 400 times. The relative frequency of green is 0.55 The number of times it landed on red is twice the number of times it landed on blue. Work out an estimate for the probability that the spinner lands on blue.

Common mistakes examiners see

  • Combining two experiments by averaging their relative frequencies. With 24 reds in 60 spins and 46 reds in 140 spins, students write (0.4+0.3286)÷2=0.364(0.4 + 0.3286) \div 2 = 0.364 instead of the correct 70200=0.35\frac{70}{200} = 0.35.

    Put the two experiments end to end and treat them as one: total successes over total trials. The averaging shortcut only happens to work when both experiments have the same number of trials, and Edexcel deliberately makes them different.

  • Writing the fraction upside down, so 9 sixes in 50 rolls becomes 509=5.56\frac{50}{9} = 5.56.

    The number of trials is always the denominator. Any relative frequency above 1 is impossible, so check the size of the answer before writing it in the box: it has to sit between 0 and 1.

  • Dividing by the number of different outcomes rather than the number of trials. A spinner with four colours spun 250 times, landing on blue 80 times, gets written as 804=20\frac{80}{4} = 20.

    Add the frequencies up first and use that total. If the question already gives the total number of trials, tick that number off in the stem so you use it rather than the count of columns.

  • Answering the 'give a reason' part without any figures, or contradicting yourself. 'The spinner is biased because it landed on D the most' names the largest column and never compares it with anything, so the mark is not awarded.

    Write the comparison as a sentence with two numbers in it: 'a fair spinner would give about 50 fours in 200 spins, and 82 were recorded, so it is biased towards 4'. Give one verdict and stick to it; a script that says biased in one line and fair in the next scores zero for both.

  • Calling a dice biased on the strength of 12 rolls, or picking the estimate from the smallest experiment because the numbers divide neatly. Four sixes in 12 rolls looks damning but is entirely ordinary.

    Judge bias only from a large number of trials, and quote the number of trials in your reason. When two estimates are offered, choose the one from more trials: that is exactly what spec point P5 says, that results settle towards the true probability as the sample grows.

Frequently asked questions

What is the difference between relative frequency and probability?
Theoretical probability comes from counting equally likely outcomes, so a fair five-sided spinner has P(5) = 1/5 before anyone spins it. Relative frequency comes from an experiment: it is the number of times the outcome happened divided by the number of trials. If the spinner is biased there is no way to count the theoretical value, so the experiment is all you have, which is why Edexcel words those questions 'work out an estimate' or 'use the results to find'.
How do you show that a spinner or a dice is biased?
Compare what happened with what a fair object would have produced, and quote both numbers. If a four-sided spinner is spun 200 times, a fair one would land on each number about 50 times, so 82 landings on one number is strong evidence of bias towards it. A reason with no figures in it, or one based on only a handful of trials, does not earn the mark.
Why do more trials give a better estimate of the probability?
Spec point P5 puts it as unbiased samples tending towards the theoretical distribution as the sample size increases. In practice, a run of luck can push 10 trials a long way from the true value, while the same run barely moves the figure over 1000 trials. So when a question offers you several estimates, the reliable one is always the one based on the most trials.
Is relative frequency on the Foundation paper too?
Yes. P2, P3 and P5 are common content, so estimating a probability from results, working out expected frequencies and judging fairness all appear on Foundation papers. On a Higher paper the question tends to sit in the first half and to be worth 2-5 marks, often with a combining-two-experiments step that Foundation questions leave out.

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