GCSEStatistics: HigherEdexcelRevision NotesProbabilityProbability BasicsRisk & Bias

Risk & Bias (Edexcel GCSE Statistics)

Revision Note

Roger

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Roger

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Maths

Relative & Absolute Risks

How do I calculate the absolute risk of an event happening?

  • You can use collected data to estimate the probability that an event will occur

    • If the event is something negative or undesirable

      • then the estimated probability is known as the risk (or absolute risk) of the event occurring

    • risk equals fraction numerator number space of space times space the space event space occurs over denominator total space number space of space trials end fraction

      • This is just the experimental probability of the 'risky' event occurring

      • So risk will always be a number between zero and one

    • e.g. an area in a particular town has flooded in 8 years out of the past 50 years

      • the risk of the town flooding in a given year is 8 over 50 equals 0.16

  • Insurance companies use risk to decide how much to charge for insurance

    • Higher risk means higher cost for the insurance

How do I calculate relative risk?

  • Relative risk measure how many times more likely an event is to occur for one group, compared with another group

    • Relative space risk space for space group space straight A space compared space with space group space straight B equals fraction numerator risk space for space group space straight A over denominator risk space for space group space straight B end fraction

      • This is not a probability

      • So relative risk does not have to be between zero and one

    • e.g. the risk of having an accident is 0.025 for people who have taken a driving safety course, and 0.09 for people who have not taken the course

      • the relative risk of having an accident for people who haven't taken the course compared with people who have taken the course is fraction numerator 0.09 over denominator 0.025 end fraction equals 3.6

      • i.e. a person who hasn't taken the course is 3.6 times more likely to have an accident than someone who has taken the course

  • Note that relative risk doesn't tell you how likely an event is in the first place

    • It only compares risk between two groups

      • e.g. relative risk may tell us that members of group A are 50 times more likely to develop a disease than members of group B

      • But if the risk for members of B is very small, then the risk for members of group A may also be small (just not as small)

Worked Example

Torbjorn recorded the number of days that it rained in his village in March and in November in 2023.

Based on this data, Torbjorn calculates that the absolute risk of it raining on a day in March is 0.323.

(a) Find the number of days that it rained in March.

Use the risk formula risk equals fraction numerator number space of space times space the space event space occurs over denominator total space number space of space trials end fraction
Put the info we know into the formula, and solve to find the info we don't know
Note that each day was a 'trial' here, and there are 31 days in March

0.323 equals fraction numerator rainy space days space in space March over denominator 31 end fraction rainy space days space in space March space equals space 0.323 cross times 31 equals 10.013

Round to the nearest whole day

10 days

Also based on his data, Torbjorn calculates that the relative risk of rainfall on a day in November compared with a day in March is 1.34.

(b) Find the number of days that it rained in November.

Use the relative risk formula Relative space risk space for space group space straight A space compared space with space group space straight B equals fraction numerator risk space for space group space straight A over denominator risk space for space group space straight B end fraction

Here 'group A' is 'days in November' and 'group B' is 'days in March'
Substitute in the info we know and find the risk for November

1.34 equals fraction numerator risk space for space November over denominator 0.323 end fraction risk space for space November space equals space 1.34 cross times 0.323 equals 0.43282

Now proceed as in part (a) to find the number of rainy days in November
Remember that there are 30 days in November

0.43282 equals fraction numerator rainy space days space in space November over denominator 30 end fraction rainy space days space in space November space equals space 0.43282 cross times 30 equals 12.9846

Round to the nearest whole day

13 days

Identifying Possible Bias

  • Estimated probabilities from data can help to identify bias

    • For example, whether or not a dice is fair

      • i.e. whether it has an equal probability of landing on each side

    • If a dice (or spinner, coin, etc.) is not fair, then it is called biased

  • Compare the expected number of times an event would occur (e.g. if the dice was fair)

    • with the actual number of times the event occurs

      • A large difference between the two suggests bias

      • A small difference may just be due to randomness

  • Remember that an estimated probability tends to get closer and closer to the exact probability as the number of trials increases

    • So a larger number of trials gives a more reliable result than a smaller number of trials

  • In addition to considering basic probabilities

    • you may be asked to compare experimental results to the expected results for a binomial distribution

    • See the 'Binomial Distribution' revision note for how to calculate binomial distribution probabilities

Exam Tip

If a question asks you how to increase the reliability of an experiment to detect bias

  • the answer will usually involve increasing the number of trials

Worked Example

Blaise suspects that a 6-sided dice he is using for a board game is biased. He decides to toss the dice 120 times and record how many times each number appears.

(a) Calculate the expected frequency for each number on the dice, assuming that the dice is fair.

If the dice is fair it has a 1 over 6 probability of landing on each side
Multiply that by 120 to find the expected frequency

120 cross times 1 over 6 equals 20

If the dice is fair, the expected frequency for each number on the dice is 20

Blaise summarises the results of his experiment in the following table:

number on dice

1

2

3

4

5

6

frequency

16

17

20

18

32

17


(b) Comment on these results, with reference to Blaise's belief that the dice is biased.

Note that most of the results are close to the expected frequency of 20
But the frequency for '5' (32) is quite a bit larger than 20
This suggests that the dice is biased

The frequencies for 1, 2, 3, 4 and 5 are all close to 20
However '5' occurred 32 times, which is a lot higher than 20
This supports Blaise's belief that the dice is biased


Blaise's friend Pierre thinks that the dice is fair, and that Blaise's results in the experiment are just the result of chance.

(c) Suggest something that Blaise might do to help convince Pierre that the dice is indeed biased.

Remember that increasing the number of trials increases the reliability of the results

Blaise could conduct another experiment, but this time roll the dice a larger number of times

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    Roger

    Author: Roger

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.