Risk (Higher) - Worksheets, Questions and Revision

16 original exam-style questions - 12 pages of questions with a full mark scheme - free printable PDF.

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HIGHER

H03 Risk (Higher)

EDEXCEL 1ST0 · Calculator allowed · about 85 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

A car parts factory states that the risk of a particular sensor failing during its warranty period is 1 in 40.

(a)Write the risk of 1 in 40 as a fraction.(1)

(b)Write the risk of 1 in 40 as a decimal.(1)

(c)Write the risk of 1 in 40 as a percentage.(1)

(d)The factory sells 6000 of these sensors during the warranty period. Work out the number of sensors expected to fail.(2)

(Total for Question 1 is 5 marks)

Which of these is equivalent to a risk of 1 in 8?

A) 8%
B) 12.5%
C) 1.8%
D) 80%

(Total for Question 2 is 1 mark)

For each statement about risk, odds and independent events, write down whether it is True or False.

(i)A risk of 1 in 500 is smaller than a risk of 1 in 25.(1)

(ii)Odds of 1 : 4 against an event happening mean the same thing as a probability of 1/4 for that event.(1)

(iii)If two risks are independent, the probability that both events happen is found by multiplying their individual probabilities together.(1)

(Total for Question 3 is 3 marks)

Three parts suppliers report the risk of a faulty part in their most recent shipments.

Supplier P: risk of a fault = 1 in 25
Supplier Q: risk of a fault = 3%
Supplier R: risk of a fault = 1/16

(a)Convert each risk to a percentage, and hence write the three suppliers in order of risk, from lowest to highest.(3)

(b)State which supplier has the safest (lowest risk) production process.(1)

(Total for Question 4 is 4 marks)

The probability that a delivery van is delayed on a particular route is 0.2.

(a)Write this probability as odds against the van being delayed, giving your answer in the form n : 1.(2)

(b)A different component in a factory has odds against failure during a shift of 7 to 3. Work out the probability that this component fails during a shift.(2)

(c)Explain why odds and probability are not exactly the same way of measuring how likely an event is.(1)

(Total for Question 5 is 5 marks)

A bookmaker quotes odds for two horses in a race.

Horse 1: odds against winning = 9 to 1
Horse 2: odds on winning = 2 to 1 (Horse 2 is more likely to win than to lose)

(a)Work out the probability that Horse 1 wins the race, as a percentage.(2)

(b)Explain the difference between odds on and odds against, and work out the probability that Horse 2 wins the race.(2)

(Total for Question 6 is 4 marks)

The bar chart shows the risk of a printing defect, as a percentage, on four machines at a print factory.

(a)Write down the risk of a defect on Machine M2.(1)

(b)Write down which machine has the lowest risk of a defect.(1)

(c)Machine M1 produces 4500 posters per day. Work out the number of defective posters expected from Machine M1 per day.(2)

(d)Machine M3 produces 12000 posters per day, and Machine M4 produces 2000 posters per day. Work out which machine is expected to produce more defective posters per day, showing your working.(3)

(Total for Question 7 is 7 marks)

An allergy clinic records data on allergic reactions in two age groups.

Group	Number in group	Number with an allergic reaction
Children	400	20
Adults	1000	10

(a)Work out the absolute risk of an allergic reaction for a child, as a percentage.(2)

(b)Work out the absolute risk of an allergic reaction for an adult, as a percentage.(2)

(c)Work out the relative risk of an allergic reaction for a child compared with an adult.(2)

(d)Explain the difference between absolute risk and relative risk, using your answers to parts a to c as an example.(1)

(Total for Question 8 is 7 marks)

A security system has two independent alarm components. Component X has an annual risk of failing to activate of 5%. Component Y has an annual risk of failing to activate of 2%.

(a)Work out the probability that both components fail to activate in a year.(2)

(b)Work out the probability that neither component fails (that is, the system definitely works as intended).(2)

(c)Explain why it is valid to multiply the two individual probabilities together in part a.(1)

(Total for Question 9 is 5 marks)

In a particular year, the risk that a certain resident's house floods is 8%, and the independent risk that the house has a gas leak is 3%.

(a)Work out the probability that the resident experiences at least one of these two hazards during the year.(3)

(b)Write this risk approximately in the form 1 in n, giving n to the nearest whole number.(2)

(c)A resident says, "The risk of at least one hazard must just be 8% + 3% = 11%." Explain why this method is not exactly correct.(1)

(Total for Question 10 is 6 marks)

A cycling club wants to investigate the claim: "Wearing the new lightweight helmet design reduces the risk of head injury compared with the standard helmet, in the event of a crash."

(a)Plan: Write a suitable question the club could use to plan this investigation.(1)

(b)Collect: The club decides to compare real crash records for two groups of cyclists, rather than testing helmets on crash-test dummies in a laboratory. Suggest one advantage of using real crash records for this investigation.(1)

(c)Process: In the study, 3000 cyclists wore the new helmet and 3000 wore the standard helmet. Of these, 12 new-helmet wearers and 30 standard-helmet wearers suffered a head injury in a crash during the year. Work out the relative risk of head injury for a standard-helmet wearer compared with a new-helmet wearer.(3)

(d)Interpret: Write a conclusion, referring to your answer to part c, that the club could use to support or reject the original claim.(2)

(Total for Question 11 is 7 marks)

A health board is comparing two possible treatments for reducing infection risk after surgery.

Treatment A: risk of a wound infection falls from 8% (without treatment) to 2% (with treatment)
Treatment B: risk of a different infection falls from 25% (without treatment) to 10% (with treatment)

(a)Work out the absolute risk reduction (ARR) for Treatment A and for Treatment B.(2)

(b)Work out the relative risk reduction (RRR) for Treatment A and for Treatment B, as percentages.(3)

(c)State, giving a reason based on your answers, which treatment reduces risk by the greater percentage of its original value, and which treatment prevents more cases per 100 patients treated.(2)

(d)Explain why relative risk reduction alone can be a misleading way to compare two treatments that treat different conditions with different original risks.(1)

(Total for Question 12 is 8 marks)

A savings and loans comparison website lists the risk of default (failing to repay) for three short-term loan products.

Loan A: odds against default = 24 to 1
Loan B: risk of default = 5%
Loan C: probability of default = 1/12

(a)Work out the probability of default for Loan A, as a percentage.(2)

(b)Work out the probability of default for Loan C, as a percentage, correct to 1 decimal place.(2)

(c)Using your answers, and the risk given for Loan B, put the three loans in order of risk of default, from lowest to highest.(2)

(d)An investor says, "Loan A is the safest because its odds against default are the biggest number." Comment on whether this is a reliable way to judge risk in general.(1)

(Total for Question 13 is 7 marks)

A fire alarm has an annual risk of failing its safety test of 1 in 20, independently from year to year.

(a)Work out the probability that the alarm passes its safety test in a single year.(1)

(b)Work out the probability that the alarm passes its safety test in every one of the next 3 years, as a percentage correct to 1 decimal place.(2)

(c)Hence work out the probability that the alarm fails its safety test in at least one of the next 3 years, as a percentage correct to 1 decimal place.(2)

(d)A fire safety company maintains 400 identical alarms, each independent of the others. Estimate how many of these alarms would be expected to fail their safety test at least once during the next 3 years.(2)

(Total for Question 14 is 7 marks)

A news headline claims: "New vaccine slashes infection risk by 80%!" In a clinical trial, 2000 people received the vaccine and a separate 2000 people received a placebo (a dummy injection). During the trial, 20 of the vaccinated group and 100 of the placebo group caught the infection.

(a)Work out the risk of infection for the vaccinated group and for the placebo group, both as percentages.(2)

(b)Work out the relative risk reduction (RRR) achieved by the vaccine, as a percentage.(2)

(c)Work out the absolute risk reduction (ARR) achieved by the vaccine.(1)

(d)Comment on whether the headline gives a complete and fair picture of the benefit to an individual, referring to both your relative and absolute risk figures.(1)

(Total for Question 15 is 6 marks)

A health study identifies two independent risk factors for developing a particular condition.

Risk factor 1 (family history): odds against developing the condition = 9 to 1
Risk factor 2 (occupational exposure): odds against developing the condition = 24 to 1

A person has both risk factors, and the two risk factors act independently of each other.

(a)Work out the probability of developing the condition from Risk factor 1 alone, and from Risk factor 2 alone, both as percentages.(2)

(b)Work out the probability that the person does not develop the condition from either risk factor.(2)

(c)Hence work out the probability that the person develops the condition as a result of at least one of the two risk factors, as a percentage.(2)

(d)Write the combined risk from part c approximately in the form 1 in n, giving n correct to 3 significant figures.(2)

(e)Comment on how the combined risk (13.6%) compares with the risk from either factor considered alone (10% and 4%), and explain why simply adding the two individual risks (10% + 4% = 14%) would not give the exact combined value.(1)

(Total for Question 16 is 9 marks)

Mark scheme · H03 Risk

Question 1

(a) B1 1/40 cao
(a) Answer: 1/40
(b) B1 0.025 cao
(b) Answer: 0.025
(c) B1 2.5% cao
(c) Answer: 2.5%
(d) M1 6000 divided by 40, oe
(d) A1 150 cao
(d) Answer: 150

Question 2

B1 B cao
Answer: B) 12.5%

Question 3

(i) B1 True cao
(i) Answer: True
(ii) B1 False cao
(ii) Answer: False
(iii) B1 True cao
(iii) Answer: True

Question 4

(a) M1 at least two of the three risks converted to percentages, oe
(a) A1 4%, 3% and 6.25% all correct
(a) A1 Q, P, R cao (correct order, lowest to highest)
(a) Answer: P = 4%, Q = 3%, R = 6.25%; order: Q, P, R
(b) B1 Q cao, ft from part a
(b) Answer: Supplier Q

Question 5

(a) M1 0.8 : 0.2, oe
(a) A1 4 : 1 cao
(a) Answer: 4 : 1
(b) M1 3 divided by (7 + 3), oe
(b) A1 0.3 or 30% cao
(b) Answer: 0.3 (30%)
(c) B1 valid explanation, e.g. probability compares the number of successful outcomes to ALL possible outcomes, whereas odds compare the successful outcomes only to the unsuccessful outcomes, oe
(c) Answer: Probability compares successful outcomes to all outcomes; odds compare successful outcomes to unsuccessful outcomes only, so the two numbers are calculated differently even though both describe likelihood.

Question 6

(a) M1 1 divided by (9 + 1), oe
(a) A1 10% cao
(a) Answer: 10%
(b) B1 correct explanation, e.g. odds against are written as (chance of losing) : (chance of winning), used when an event is less likely than not; odds on are written the other way round, (chance of winning) : (chance of losing), used when the event is more likely than not, oe
(b) B1 2/3 or 66.7% awrt cao
(b) Answer: 2/3 (approximately 66.7%)

Question 7

(a) B1 5% cao
(a) Answer: 5%
(b) B1 M3 cao
(b) Answer: M3
(c) M1 4500 x 0.02, oe
(c) A1 90 cao
(c) Answer: 90
(d) M1 12000 x 0.008, oe (= 96)
(d) M1 2000 x 0.03, oe (= 60)
(d) A1 M3 cao, with 96 and 60 both stated
(d) Answer: M3 (96 defective posters per day, compared with 60 for M4)

Question 8

(a) M1 20 divided by 400, oe
(a) A1 5% cao
(a) Answer: 5%
(b) M1 10 divided by 1000, oe
(b) A1 1% cao
(b) Answer: 1%
(c) M1 5% divided by 1%, oe
(c) A1 5 (times) cao
(c) Answer: 5 times
(d) B1 valid explanation, e.g. absolute risk is the actual chance of the event happening to one person in a group (5% for children, 1% for adults), while relative risk compares two absolute risks to show how many times more or less likely one group is (children are 5 times as likely as adults), oe
(d) Answer: Absolute risk is the actual chance for one person in a group (5% for children, 1% for adults); relative risk compares the two absolute risks (5% divided by 1% = 5), showing children are 5 times as likely to react.

Question 9

(a) M1 0.05 x 0.02, oe
(a) A1 0.1% or 0.001 cao
(a) Answer: 0.1%
(b) M1 0.95 x 0.98, oe
(b) A1 93.1% cao
(b) Answer: 93.1%
(c) B1 valid explanation referring to independence, e.g. the two components are independent, so Component X failing does not change the probability of Component Y failing, meaning the individual probabilities can be multiplied, oe
(c) Answer: The components are independent, so one failing does not affect the probability of the other failing, meaning their probabilities can be multiplied together.

Question 10

(a) M1 0.92 x 0.97, oe (= 0.8924)
(a) dM1 1 - their 0.8924, oe (dependent on the first method mark)
(a) A1 10.76% cao
(a) Answer: 10.76%
(b) M1 1 divided by 0.1076, oe (or ft from part a)
(b) A1 9 cao (awrt)
(b) Answer: approximately 1 in 9
(c) B1 valid explanation, e.g. simply adding the risks double-counts the (small) chance that both hazards happen in the same year, so the correct method (1 minus the product of both 'does not happen' probabilities) gives the slightly lower value of 10.76% rather than 11%, oe
(c) Answer: Adding the risks double-counts the chance both hazards happen together; the correct combined risk is 10.76%, slightly less than 11%.

Question 11

(a) B1 any suitable comparative question linking helmet type and head injury risk, e.g. 'Do cyclists wearing the new lightweight helmet have a lower risk of head injury in a crash than cyclists wearing the standard helmet?', oe
(a) Answer: e.g. Do cyclists wearing the new lightweight helmet have a lower risk of head injury in a crash than cyclists wearing the standard helmet?
(b) B1 any suitable advantage, e.g. real crash records reflect genuine crash conditions and human factors, giving a more realistic estimate of risk than a controlled dummy test, oe
(b) Answer: e.g. Real crash records capture genuine crash conditions and behaviour, giving a more realistic estimate of risk than a controlled laboratory test.
(c) M1 0.4% and 1% found, oe (12/3000 and 30/3000 converted to percentages)
(c) dM1 1% divided by 0.4%, oe (dependent on the first method mark)
(c) A1 2.5 (times) cao
(c) Answer: 2.5 times
(d) B1 conclusion consistent with the data supporting the claim
(d) B1 correctly references the relative risk figure from part c, e.g. 'This supports the claim: standard-helmet wearers are 2.5 times as likely to suffer a head injury as new-helmet wearers, so the new helmet does appear to reduce the risk of head injury', oe
(d) Answer: e.g. This supports the claim: standard-helmet wearers are 2.5 times as likely to suffer a head injury as new-helmet wearers, so the new helmet does appear to reduce the risk of head injury.

Question 12

(a) M1 8 - 2 or 25 - 10, oe (method for at least one)
(a) A1 6 percentage points and 15 percentage points, both correct
(a) Answer: Treatment A: 6 percentage points; Treatment B: 15 percentage points
(b) M1 6 divided by 8 x 100 or 15 divided by 25 x 100, oe (method for at least one)
(b) A1 75% cao (Treatment A)
(b) A1 60% cao (Treatment B)
(b) Answer: Treatment A: 75%; Treatment B: 60%
(c) B1 Treatment A has the greater relative risk reduction (75% versus 60%)
(c) B1 Treatment B prevents more cases per 100 patients treated (15 versus 6, from the absolute risk reduction)
(c) Answer: Treatment A has the greater relative reduction (75% vs 60%); Treatment B prevents more cases per 100 patients (15 vs 6).
(d) B1 valid explanation, e.g. RRR depends on the size of the original risk, so a large percentage reduction of a small risk can prevent fewer cases in real terms than a smaller percentage reduction of a larger risk; the absolute risk reduction (or actual number of cases prevented) also needs to be considered, oe
(d) Answer: RRR depends on the size of the original risk, so a high RRR does not always mean more cases are prevented in real terms; ARR (or actual case numbers) should also be considered.

Question 13

(a) M1 1 divided by (24 + 1), oe
(a) A1 4% cao
(a) Answer: 4%
(b) M1 1 divided by 12, oe
(b) A1 8.3% awrt
(b) Answer: 8.3%
(c) M1 all three risks compared on a common scale (4%, 5%, 8.3%), oe
(c) A1 A, B, C cao (in that order)
(c) Answer: A, B, C
(d) B1 valid comment, e.g. odds must be converted to a common measure (such as probability or percentage) before comparing across different products, since risks are not always given in odds form, as shown by Loans B and C here, oe
(d) Answer: Not fully reliable in general - odds need converting to a common measure (like percentage) before comparing, since other products may be given as a percentage or fraction instead of odds, as here.

Question 14

(a) B1 0.95 or 95% cao
(a) Answer: 0.95 (95%)
(b) M1 0.95 cubed, oe
(b) A1 85.7% awrt
(b) Answer: 85.7%
(c) dM1 1 - their 0.857375, oe (dependent on part b)
(c) A1 14.3% awrt
(c) Answer: 14.3%
(d) M1 400 x 0.142625, oe (or ft from part c)
(d) A1 57 cao (awrt)
(d) Answer: 57

Question 15

(a) M1 20 divided by 2000 or 100 divided by 2000, oe (method for at least one)
(a) A1 1% and 5%, both correct
(a) Answer: Vaccinated: 1%; Placebo: 5%
(b) M1 (5 - 1) divided by 5 x 100, oe
(b) A1 80% cao
(b) Answer: 80%
(c) B1 4 percentage points cao
(c) Answer: 4 percentage points
(d) B1 valid comment, e.g. the 80% figure (RRR) is mathematically correct, but on its own it can overstate the real-world benefit, since the absolute risk only falls from 5% to 1%, a drop of just 4 percentage points, so most people in either group would not have caught the infection anyway, oe
(d) Answer: The 80% RRR is correct but can overstate the benefit on its own, since the absolute risk only falls from 5% to 1% (4 percentage points), so most people in both groups would not have been infected anyway.