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P1 Using a combination of statistics to investigate risk and its impact on life

In this chapter you will learn how to: • develop the link between simple probability and expected frequency • explore different contexts in which probability is used, such as risk assessment.

You should already know how to: • calculate simple probability • use probability to make predictions • convert between fractions, decimal fractions and percentages.

Probability Probability is the chance of an event occurring. It is always expressed as a number between 0 and 1. A probability scale is shown below.

0

impossible

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

even chance certain

We use the symbol P(event) for the probability that an event will occur. It is calculated like this:

P(event) = number of favourable outcomes

total number of outcomes

Probability can be expressed as a fraction, a percentage or a decimal fraction. For example:

• fractions are used in making initial calculations and tend to be used in dice and card games

• percentages tend to be used when working with weather and other geographical concepts

• decimal fractions are used by psychologists and scientists for ease of use in analysis and in further calculation, and for making comparisons.

One side of a dice is a 6.

There are six possibilities: {1, 2, 3, 4, 5, 6}

P(6) = 1

6

Example P1.1 A fair dice is rolled. What is the probability of rolling a 6?

Always write the fraction in its simplest form. Avoid changing your fraction into a percentage or decimal fraction if it needs to be rounded as this can give you an inaccurate answer.

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Expected frequency (or value)The expected frequency or number of favourable outcomes can be calculated as follows:

Expected number=probability×number of trials

Expected days of rain = 70% of 30

= 0.7 × 30

= 21 days

Example P1.2The probability of rain in Inverness in June is 70%.

How many days of rain would you expect next year in June?

This answer gives an exact number of days. If the answer does not give a whole number, make sure that you round appropriately depending on the context of the question.

Relative frequencyThe relative frequency of an event happening is a way of measuring how often an event happens, and is expressed in ways that make it possible to make comparisons. Relative frequency is based on experimental data and is an estimate of frequency.

Relative frequency number of favourable tri== aalstotalnumber of trials

If an experiment is carried out the results can be recorded in a frequency table.

If an experiment is based on a theoretical probability, the relative frequency will eventually provide the same result as the theoretical probability, so increasing the number of trials gives a more accurate estimate for probability.

Example P1.3A coin is tossed 50 times. The results are shown in the table.

Outcome head tail

Frequency 28 22

a What is the relative frequency of tossing a coin and getting a head?

b Using the results from the table, how many times would you expect to get each result if the coin was tossed 500 times?

c How many times would you expect to get each outcome in the original trial if the coin was fair? What does this tell you about the coin?

(continued )

a P(head) = =2850

1425

b Expected number of heads = × =1425 500 280

Expected number of tails = 500 - 280 = 220

Since there are only two possible outcomes (heads or tails) you only need to work out one probability.

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Exercise P1A1 A five- sided spinner is used in a game and the score on the side which

it lands on is recorded.

a What is the probability (as a fraction) that the score is:

i 3 ii odd iii less than 5 iv prime v 6?

b Express each probability in a as a decimal fraction.

c Express each probability in a as a percentage.

d If the spinner is spun 120 times, how many times would you expect to get a score of 5?

2 A dice is rolled 60 times. The results are shown in the table.

5

1

23

4

c Based on equally likely outcomes:

P(head) P(tail)= = 12

So the expected number of heads = × =12 50 25

and the expected number of tails = × =12 50 25

The actual numbers of heads and tails (28 and 22) are different from the expected numbers of 25 each. This may mean that the coin is not fair and is weighted towards the head. But the initial trial might be too small (28 and 22 are only 3 either side of 25).

A larger trial is needed to confirm if the coin is fair or not.

Number on dice 1 2 3 4 5 6

Frequency 8 9 7 18 20 8

a What is the relative frequency of rolling a 6?

b Using the results from the table, how many times would you expect to get each number if the dice was rolled 600 times?

c How many times would you expect to get each outcome if the dice was fair?

d Compare your answers to b and c.

What does this tell you about the dice?

Age range (years) 0– 9 10– 19 20– 29 30– 39 40– 49 50– 59 60– 69 70– 79

Number of people admitted to hospital

36 21 35 13 10 5 5 3

3 Sunburn affects people of all ages. In a study by the NHS in 2009 the following results show the number of people admitted to hospital after being burnt.

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a What is the relative frequency that a person admitted to hospital for sunburn is:

i aged 0- 9 years ii aged 50- 59 years?

b There were 21 people aged between 10 and 19 who required hospital treatment. Of the 21, 18 were female. What is the relative frequency that a person admitted to hospital for sunburn is:

i male aged 10- 19 years

ii female aged 10- 19 years?

c Look at your answers to a and b.

i Can you think of reasons why the 0- 9 age range and the 20- 29 age range are higher than other age ranges?

ii Why do you think fewer people are admitted to hospital as they get older?

4 In a study carried out by the Food Standards Agency of Scotland in 2005 it was discovered that, on average, 17% of total calories consumed by children came from sugar.

a If the average calorie intake for a child is 1800 calories, what is the expected number of calories consumed per child from sugar?

b Too much sugar in a child’s diet can have lasting negative effects. There is a risk of tooth decay, obesity and in extreme cases there is a risk of type 2 diabetes.

If there are 900 000 children and teenagers under the age of 16, how many are likely to develop:

i tooth decay, if the probability of developing tooth decay is 13

ii obesity, if the probability that a person becoming obese or morbidly obese is 14.4%

iii type 2 diabetes, if the probability of a person developing type 2 diabetes by the age of 65 is 0.05?

c The Scottish dietary target is for a maximum of 10% of total calories consumed by children to come from sugar. What recommendations would you give to the Government to help reach this figure and how do you think this could be implemented?

5 The table shows the percentage of people consuming five or more portions of fruits and vegetables in 2001 and 2007 in the UK.

In part a remember to add the frequencies together to give the total for the denominator of your fraction.

Men Women Children ( 2– 15 years)

2001 22 25 10

2007 27 31 21

a Using the data for children aged 2–15, if there were 1200 pupils in a school in 2001, how many pupils would you expect to have regularly consumed five or more portions of fruits and vegetables?

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b Which category (men, women or children) is best at consuming their five- a- day? Why do you think this is the case?

c Which category (men, women or children) showed the biggest increase in percentage of people consuming five or more portions of fruits and vegetables?

Risk and probabilityRisk is the potential that a chosen activity or action may result in a loss or undesired outcome and can be described as the probability of uncertain future events. Risks are taken every day in sport or other leisure activities, driving a car, cooking, investing money, and so on. Predictions of risk are determined by known factors and statistics of past events.

Managing risk is an important factor in areas such as health and safety, insurance, finance and security.

Exercise P1B1 The table shows the chances of dying from various causes in a year in the UK.

Cause of death Probability

Cancer 1 in 360

External causes 1 in 3070

Road accidents 1 in 15 700

Being struck by lightning 1 in 15 000 000

The population of Scotland is 5.2 million.

a Approximately how many people would you expect to die each year from each of the causes above?

b How many times more likely is it that a person will die from cancer than from a road accident?

Death type Teen motor vehicle deaths

Driver 2411

Passenger 2017

Pedestrian 326

Motorcyclist 237

Bicyclist 100

Other 99

Total 5190

2 A total of 5190 teenagers died in motor vehicle crashes in the US in 2005.

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a What is the relative frequency of each type of death?

b What is the relative frequency of death in a motor vehicle crash for teenagers by being a driver or passenger?

c i The total number of motor vehicle deaths for all ages in 2005 was 43 443. What is the relative frequency of teenage deaths?

ii Use the relative frequency in c i as the probability that a motor death is a teenager. In New York State there were 1900 motor vehicle deaths. How many of these would you expect to be teenagers?

d What recommendations or policies could the US Government make to reduce the number of driver and passenger deaths?

3 The Health and Safety Executive (HSE) aim to reduce the number of injuries and fatalities in the workplace in the UK.

a Their results in 2011 revealed that the relative frequency of a non- fatal accident in the workplace in the UK was 0.004 454. If the working population of the UK was approximately 33 million, estimate how many people had a work- related non- fatal accident in 2011.

b i 10% of all non- fatal injuries took place in the construction industry. Calculate how many of the injuries calculated in a took place in the construction industry.

ii 5% of the total workforce worked in the construction industry. How many people worked in the construction industry?

iii What is the probability (expressed as a decimal fraction) that a person who worked in the construction industry had a non- fatal accident?

iv What could management in the construction industry do to minimise risk to their workforce?

c i 7% of all non- fatal injuries took place in the health and social care industry. Calculate how many of the injuries calculated in a took place in the health and social care industry.

ii 8% of the total workforce worked in the health and social care industry. How many people worked in the health and social care industry?

iii What is the probability (expressed as a decimal fraction) that a person who worked in the health and social care industry had a non- fatal accident?

iv What could management in the health and social care industry do to minimise risk to their workforce?

v Based on your results to parts b iii and c iii, which is more of a risk: working in the health and social care industry or the construction industry? Give a reason for your answer.

d The number of fatalities in the workplace in the UK was 165.

i Using the UK working population as 33 million, calculate the relative frequency of a fatality in the workplace in the UK in 2011.

ii The working population of Scotland in 2011 was approximately 2 450 000. Using your answer to d i as the probability of a fatal accident, estimate the number of workplace fatalities in Scotland in 2011.

In part b add the passenger and driver figures together.

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iii In Scotland 34 cases were prosecuted for health and safety breaches in 2011. This was a drop of 11% on the previous year. How many cases were heard in 2010?

4 Based on crime statistics in England and Wales, the table shows vehicle- related theft reported during 2005- 2011.

Year 2005 2006 2007 2008 2009 2010 2011

Total vehicle- related theft 1673 1635 1458 1459 1213 1189 1217

Theft from vehicles 1080 1092 968 1021 839 858 894

Theft of vehicles 180 171 157 144 113 94 94

Attempts of theft of and from vehicles

413 372 333 294 261 237 229

a What is the relative frequency of a vehicle- related crime reported in 2009 being reported as a theft of a vehicle?

b Describe the trend in vehicle- related crime from 2005 to 2011.

c What factors contribute to this trend?

d What impact does vehicle- related theft have on insurance?

Risk assessmentRisk assessment is the process of analysing an activity or event to forecast potential risks and to then take steps to reduce possible exposure to risk. In most cases in everyday life, the probability of a risk is not significant. The second element of a risk assessment is a consideration of the severity of the potential hazard.

For example, if you are comparing travel by road or by air, the probability of a road accident is higher but the severity of the hazard in travelling by air is much greater.

A scale of risk is shown in the diagram below, showing the probability (calculated between 0 and 1) and the severity of the hazard (calculated between 0 and 100). (0 represents ‘No hazard’ and 100 represents ‘Severe hazard’.)

0

1

1000

Prob

abili

tyof

the

even

t

Hazard ofthe event

Lowrisk

Mediumrisk

Mediumrisk

Mediumrisk

Highrisk

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Activity 1 - Outcome B: high probability and high risk. There is a high chance of someone drinking in excess getting a liver- related disease and if they do they are likely to die from it.

Activity 2 - Outcome E: low probability and high hazard. The likelihood of being struck by lightning is very low but the hazard is very serious and may result in death.

Activity 3 - Outcome A: high probability and low risk. They are very likely to catch a cold but the hazard is not serious.

Activity 4 - Outcome C: medium probability and medium risk. It is likely a person will hit their hand and it is likely to be quite painful but not life threatening.

Activity 5 - Outcome D: low probability and low hazard. It is uncommon for power cuts to spoil all the food in a freezer in the UK and, if it does happen, it is a low hazard to those involved.

Example P1.4The grid shows the risk factors of five events: A, B, C, D, E. Associate these five activities with each event on the grid:

1 a person drinking at least 5 units of alcohol a day getting a liver- related illness

2 a person carrying an umbrella being struck by lightning

3 a person working in an office sitting next to someone with a cold catching a cold as well

4 a person hitting their hand whilst hammering a tent peg into the ground

5 a power cut causing all the food in a freezer to be spoilt.

0

1

1000

Prob

abili

tyof

the

even

t

Hazard ofthe event

D E

C

A B

ActivityMake a copy of the probability– hazard diagram.

Think of different activities (similar to those in Example P1.4) and their associated likelihood and hazards.

Write down at least two examples of your own for each type of activity and mark them on your probability–hazard diagram:

• A: high probability and low hazard• B: high probability and high hazard• C: medium probability and medium hazard• D: low probability and low hazard• E: low probability and high hazard.

Compare these with others in your class.

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Make the linkWriting a risk assessmentExercise P1B Question 3 b iv and c iv asked what management in the construction and health and social care industries could do to minimise risk to their workforce.

Companies and organisations are required by law to write a risk assessment for their day- to- day activities. This is a careful examination of what could cause harm to people in the workplace or away from their workplace but in the course of doing their job, so that they can weigh up whether they have taken precautions or should do more to prevent harm. In order to write a suitable risk assessment they need to:

• identify hazards

• decide when and how accidents are likely to occur

• evaluate the risk

• record their findings and implement any necessary changes

• review and update if and when it is necessary.

Make the linkCar insuranceIn Exercise P1B Question 4 the impact of car theft and insurance is discussed. All drivers must have car insurance. If there is an increase in car theft then insurance premiums (the amount paid per month or year) will increase as there will be more people needing compensation from their insurance company.

Car insurance is partly determined by postcode as different areas have different levels of risk of car theft and accidental damage.

Other contributing factors to individual car insurance premiums include the make and model of car, mileage per year, age of driver, number of years with no claim or accident. All of these factors contribute to the risk of theft or accidental damage, whether it be the fault of the driver or a third party.

In the Equality Act of 2012, insurance companies were told it was a legal requirement (due to EU legislation) that a person’s insurance premium could not depend on their gender, even though statistics show that it is less likely that women have accidents (and are therefore less likely to make a claim to the insurance company). Many women have found that their insurance premiums have risen significantly since 2012.

Comment on the Government’s decision. Do you think their decision is fair?

Make the linkHealth and dietIn Exercise P1A Question 4 you were asked to consider recommendations you would make to the Government. The Food Standards Agency followed up their study with this recommendation:

Avoid sugary drink. Limit consumption of energy- dense foods (particularly processed foods high in added sugar, or low in fibre, or high in fat).

The full study mentioned in Exercise P1A Question 5 showed that the increase in consumption of fruit and vegetables was a direct result of government initiatives and recommendations to local authorities in heath promotion. School meal provision was improved as a result, much of which is detailed in The Schools (Health Promotion and Nutrition) (Scotland) Act 2007.

The study used probability to predict outcomes and to make recommendations to reduce future health risks.

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• I can develop the link between simple probability and expected frequency. Exercise P1A Q2

• I can explore different contexts in which probability is used, for example, considering risk. Exercise P1B Q4

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