OxfordAQA International GCSE Mathematics: Probability scale, relative and expected frequency (9260)

This study guide covers the first part of probability in the OxfordAQA International GCSE Mathematics specification (9260, Version 5.1, for exams May/June 2018 onwards): statements S9 to S13 in section 3.4.3. All five statements are Core content, so they can be assessed on both the Core and the Extension papers. There is no Extension only content in this unit.

Sample spaces, mutually exclusive events and Venn diagrams are taught elsewhere, as are independent events and tree diagrams. This page stays with the scale, estimates from data and the effect of sample size. For a shorter recap go to the revision notes. To test yourself use the practice set. The course hub is at /boards/oxfordaqa/igcse/mathematics/, the printable list of statements is at /checklists/oxfordaqa/igcse/mathematics/, and the free 10-minute diagnostics help you find a starting point.

Scope of this unit

Ref	What you must be able to do	Tier
S9	Understand and use the vocabulary of probability and the probability scale	Core
S10	Use estimates or measures of probability from theoretical models (including equally likely outcomes) or from relative frequency; use expected frequency	Core
S11	Compare experimental data and theoretical probabilities	Core
S12	Understand that repeating an experiment may, and usually will, give different outcomes	Core
S13	Understand that increasing sample size generally leads to better estimates of probability and population characteristics	Core

S9: vocabulary and the probability scale

An experiment is any activity whose result cannot be known in advance, such as spinning a spinner. Each possible result is an outcome. An event is one outcome or a group of outcomes you are interested in, such as “an even number”. A trial is one go at the experiment.

Probability measures how likely an event is. It is written as a fraction, a decimal or a percentage, and it always lies from 0 to 1 (or 0% to 100%).

0              0.5              1
|---------------|---------------|
impossible   evens (even chance)   certain
   unlikely  <---------->  likely

Probability	Word	Example
0	impossible	rolling a 7 on a normal die
close to 0	unlikely	picking the single red counter from a bag of 50
0.5	evens, or an even chance	a fair coin landing heads
close to 1	likely	not picking that red counter
1	certain	rolling a number less than 10 on a normal die

A probability can never be negative and can never be more than 1. If your working gives 1.4 or -0.2, there is a mistake.

Fair (or unbiased) means every outcome has the same chance. A biased object favours some outcomes.

Worked example 1

Put these on a probability scale and give each a word: (a) a month chosen at random has 31 days, (b) a day chosen at random is a Tuesday, (c) a randomly chosen day of the week has a name ending in “day”.

(a) Seven of the twelve months have 31 days, so the probability is 7/12. That is a little over 0.5: likely. (b) One day in seven: 1/7, about 0.14. That is unlikely. (c) Every day name ends in “day”: probability 1, certain.

S10: theoretical probability

When all outcomes are equally likely:

P(event) = favourable outcomes / all possible outcomes

The words “at random”, “fair” and “unbiased” tell you outcomes are equally likely. This is a theoretical model: you calculate the probability from the set-up, without doing any experiment.

Worked example 2

A fair spinner has 8 equal sectors. Three are shaded.

1. Outcomes in the event: 3. Total outcomes: 8.
2. P(shaded) = 3/8 = 0.375 = 37.5%.

Using a probability to find a missing number

A bag holds 12 counters. P(blue) = 1/4. The number of blue counters is 1/4 × 12 = 3. Always check that your count of favourable outcomes is a whole number.

S10: relative frequency

If outcomes are not equally likely, or you cannot work out the model, you collect data. The relative frequency of an event is:

relative frequency = number of times the event happened / total number of trials

Use relative frequency as an estimate of the probability. It is also called experimental probability.

Worked example 3

A drawing pin is dropped 200 times. It lands point up 74 times.

1. Relative frequency of point up = 74/200 = 0.37.
2. The best estimate of P(point up) from this data is 0.37.

You cannot use 1/2 here, because the two ways of landing are not equally likely. The pin is not symmetrical.

S10: expected frequency

Expected frequency is how many times you expect an event to happen in a number of trials:

expected frequency = probability × number of trials

The probability can be theoretical or an estimate from relative frequency.

Worked example 4

(a) The spinner from example 2 is spun 240 times. Expected number of shaded results = 3/8 × 240 = 90.

(b) The drawing pin from example 3 is dropped 1500 times. Estimated expected number of point-up results = 0.37 × 1500 = 555.

The word “expected” does not mean “will happen”. The expected frequency is a prediction of the average result. In (a), 88 or 93 shaded results would be quite ordinary. The expected value does not need to be a whole number either: 112.5 is a sensible expected frequency, even though you cannot observe half a result.

Working backwards

If a machine is expected to produce 45 faulty items in 150, the estimated probability of a fault is 45/150 = 0.3.

S11: comparing experimental and theoretical probabilities

Compare the relative frequency from an experiment with the theoretical probability. If they are close, the data agrees with the model. If they are far apart, there are two possible reasons: chance, or a model that is wrong (for example, a biased die).

Worked example 5

A die is rolled 120 times and a six appears 27 times. Is the die biased?

1. Theoretical P(six) for a fair die = 1/6 ≈ 0.167.
2. Relative frequency = 27/120 = 0.225.
3. Expected number of sixes = 120 × 1/6 = 20. Observed: 27, which is 7 more.
4. The data is higher than expected, but 120 rolls is a small sample, and a difference of 7 could be chance. This does not prove the die is biased.
5. To investigate further, roll the die many more times and see whether the relative frequency stays near 0.225 or settles near 0.167.

Notice step 4. In a written answer, avoid saying “the die is definitely biased” or “the die is definitely fair” from one small experiment. Say what the numbers show, then say what extra evidence you would collect.

S12: repeating an experiment

If you repeat an experiment, you will usually get different results. Roll a fair die 60 times and you might see 8 sixes. Do it again and you might see 13. Neither result is wrong.

This is why an experiment gives an estimate, not the exact probability, and why two people with identical equipment can report different relative frequencies. The differences come from chance. They do not mean one person made an error.

S13: bigger samples give better estimates

Relative frequency from a larger number of trials is generally closer to the true probability. Here is data from a fair coin, with the relative frequency of heads calculated at each stage:

Trials	Heads	Relative frequency
10	7	0.7
50	22	0.44
100	54	0.54
500	243	0.486
1000	505	0.505

The early values swing widely. By 1000 trials the value is close to the theoretical 0.5. Note the word “generally”: a larger sample does not guarantee a better result every time, but it makes a good result much more likely. A relative frequency also never needs to match the theoretical value exactly.

Samples and populations

The same idea applies when you survey part of a population (everyone you are interested in) and use the sample to estimate a population characteristic, such as the proportion who walk to school.

Worked example 6

A school has 1200 students. A first survey of 40 students finds 9 walk to school. A second survey of 200 students finds 62 walk.

1. First estimate: 9/40 = 0.225, so 0.225 × 1200 = 270 students.
2. Second estimate: 62/200 = 0.31, so 0.31 × 1200 = 372 students.
3. The second estimate is more reliable because it uses a larger sample. (This assumes both samples were chosen fairly, with no bias in who was asked.)

Size is not the only issue. A sample of 200 taken only from the students who arrive by bus would still be a poor sample. A larger sample helps only if it is chosen fairly.

Common errors

• Giving a probability greater than 1, or as a ratio such as 3:5. Use a fraction, decimal or percentage.
• Using 1/2 for every two-outcome event. That works only if the outcomes are equally likely.
• Writing relative frequency as “successes/failures” instead of “successes/total trials”.
• Calculating expected frequency as probability + trials, or dividing instead of multiplying.
• Saying the result “must” equal the expected frequency.
• Calling a die biased because one short experiment did not match the model.
• Writing “the more trials, the closer it gets, so 1000 trials are always exact”. It is an estimate, however large the sample.
• Rounding too early. Keep the exact fraction until the end, and answer to the accuracy asked.

Next steps

Work through the practice questions, then use the revision notes as a final check. Tick off S9 to S13 on the checklist.

Official syllabus

This page follows the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1, for exams May/June 2018 onwards, published by OxfordAQA, section 3.4.3 Probability, statements S9 to S13.