Project-based lesson • Maths • Ages 16–18 • Conditional probability

What does road risk cost?

The big question: “Is it fair for the price of car insurance to depend only on age – or does behaviour on the road tell us more?” Take on the role of the board of directors of an insurance company and solve the case using conditional probability.

All the numbers, client names and the company “Alpha Safe” in this lesson are fictional data created for teaching. They are not official road-safety or police statistics, and they are not data from a real insurance company. The differences between the age groups are also invented for teaching purposes.

80 minutes (2 periods)

6 parts of the lesson

P(A|B) conditional probability

Client A Age 19 · group 18–24

2 violations (speeding) · city driving

Client B Age 36 · group 25–44

0 violations · mixed driving

Client C Age 58 · group 45+

1 violation (unsafe overtaking) · out-of-town driving

Step 1

Facing realistic cases

Imagine you are managers at the insurance company “Alpha Safe” (a fictional company used as a teaching example). Three clients from different age groups come to you for car insurance.

Client A

Age: 19 (group 18–24)
Driving licence held for: 1 year
Traffic violations: 2 (speeding)
Driving: busy city traffic

Client B

Age: 36 (group 25–44)
Driving licence held for: 15 years
Traffic violations: 0 (none)
Driving: mixed

Client C

Age: 58 (group 45+)
Driving licence held for: 35 years
Traffic violations: 1 (unsafe overtaking)
Driving: mostly out of town

Not graded · your intuition only

Task 1. Your first choice

Make a quick team bet without calculating anything. What should the price of insurance for the three clients mainly depend on?

Option 1: On age – the youngest (Client A) pays the most and the oldest (Client C) pays the least. Option 2: On violations – the clients with violations (A and C) pay more than Client B, regardless of age. Option 3: On none of these – all three should pay the same basic price.

Write one sentence explaining why you chose this option: 0 characters

Step 2

Explore the teaching database

To make a fair business decision, you open Alpha Safe’s teaching database of 1,000 drivers from three age groups. For each driver, three things are recorded: their age group, whether they have traffic violations and whether they caused a road crash in the last year.

The database of 1,000 drivers is a fictional example created for teaching, built only for this lesson – it does not come from a real institution or insurance company. The numbers by age group are not real statistics and do not show real links between age and risk.

Alpha Safe teaching database: 1,000 drivers by age group, violations and crashes (fictional data)
Age group	With violations		Without violations		Group total
Age group	drivers	of whom caused a crash	drivers	of whom caused a crash	drivers	of whom caused a crash
18–24	100	30	150	15	250	45
25–44	150	30	350	21	500	51
45+	50	6	200	13	250	19
Total	300	66	700	49	1,000	115

How to read the table: each cell is a number of drivers. For example, in the “25–44” row under “With violations · drivers” there is 150 – that is how many drivers aged 25–44 have violations. The database has no data on driving experience or type of driving – only on age group, violations and crashes.

Task 2. Choose between two statements

Look at the “Total” row. Drivers with violations caused 66 crashes, and drivers without violations caused 49. Why is it misleading to compare the numbers of crashes directly (66 vs 49)? Choose the correct argument:

A) Because the total number of drivers in the two groups (300 and 700) is very different, so the comparison is unfair. B) Because the group without violations had fewer incidents in total.

Step 3

Calculating the risk mathematically

Now we use conditional probability to turn a feeling of danger into an exact number. This time nobody gives you a ready-made fraction – you build it yourselves from the teaching database in Step 2.

The conditional probability P(A|B) gives the chance that event A happens, GIVEN THAT we already know B is true. The group after the bar “|” is the conditioning group – we only look inside it:
P(A|B) = (number of cases in B where A is true) / (number of all cases in B)

Task 3. P(crash | violation)

What is the probability that a randomly chosen driver from the database caused a crash, if we know that they have traffic violations? Work step by step – the next field unlocks when the previous one is correct.

Task 4. P(crash | no violation)

What is the probability that a randomly chosen driver caused a crash, if we know that they have no traffic violations?

Task 5. How many times higher is the risk?

Use your results from Tasks 3 and 4. How many times greater is P(crash | violation) than P(crash | no violation)?

Ratio of the two conditional probabilities:

times

You can enter the exact number (to two decimal places) or round it to a whole number.

Task 6. Reversing the condition: P(violation | crash)

Now the question is reversed: if we know that a driver caused a crash, what is the probability that they have traffic violations? Careful: the event and the condition have swapped places.

Task 7. Why are the results different?

Compare the results of Tasks 3 and 6. Choose the correct explanation:

A) The numerator is different – in one task we count the drivers who caused a crash, and in the other the drivers with violations. B) When the condition is reversed, the conditioning group changes, and with it the denominator – that is why the result changes too. C) The two probabilities should actually be equal – one of the tasks contains a mistake.

Task 8. Interpretation: does the number prove a cause?

A board member says: “P(crash | violation) is considerably greater than P(crash | no violation). So it has been proved that violations cause crashes.” Which statement is the most accurate?

A) Yes – since the conditional probability is considerably higher, causation has been proved. B) The data show an association between violations and crashes in the teaching sample, but on their own they do not prove that violations cause crashes. C) There is no link at all between violations and crashes, because most drivers with violations did not cause a crash.

Step 4

Age, violations and the client’s profile

So far we have compared drivers only by violations. The teaching database also has a second characteristic – the age group. When the condition contains two characteristics at once, the conditioning group gets smaller.

Reminder: the differences between the age groups in this database are invented for teaching and do not describe real risks by age.

Task 9. P(crash | aged 18–24)

What is the probability that a randomly chosen driver caused a crash, if we only know that they are aged 18–24 (without knowing whether they have violations)?

Task 10. Profile risk of the three clients

For each client, find the conditional probability of a crash among the drivers in the database with the same profile – the same age group and the same violation status. Look at the client cards in Step 1.

A Client A (age 19, 2 violations)

B Client B (age 36, no violations)

C Client C (age 58, 1 violation)

Task 11. What does age show?

Compare the results of Tasks 9 and 10. Which statement is supported by the teaching sample?

A) Age alone determines the risk – every older driver has a lower risk than every younger driver. B) In this teaching sample the risk differs by age group and by whether there are violations – for example, Client C is the oldest, but in the sample their profile has a higher probability of a crash than Client B’s profile. C) After the age of 45, violations make no difference to the risk.

Transfer task · new fictional numbers

Task 12. The phone at the wheel

In another teaching (fictional) sample of 800 drivers, 200 used a phone while driving and 600 did not. Of those who used a phone, 34 caused a crash, and of those who did not use a phone, 51 caused a crash. The insurer is investigating a new crash and asks: what is the probability that the driver used a phone, if we know that they caused a crash?

Step 5

Final product: the insurance card

As the board of directors, complete the company’s official business document. Set a risk category and an annual price for each client, based on the profile risk from Step 4.

The prices are your free choice – there is no single correct amount. The system only shows the price ratios between the clients and compares them with the risk ratios. This is not a check of “right” or “wrong”.

Client A

18–24 · with violations

30% profile crash risk

Risk category

Annual price (€)

Client B

25–44 · without violations

6% profile crash risk

Risk category

Annual price (€)

Client C

45+ · with violations

12% profile crash risk

Risk category

Annual price (€)

Compare the price ratios and the risk ratios

The profile risks are in the ratio A : B : C = 30% : 6% : 12% = 5 : 1 : 2. Check how the prices you have set compare.

Not graded · open argument

Task 13. Business argument

Justify your prices to the insurance board: why is it fair, in market terms, for the three clients to pay different amounts? What counts more in your decision – age or violations?

Your argument: 0 characters

Step 6

Exit card

Summarise what you have learned before you defend your card in front of the “insurance board” (the class).

Not graded · open reflection

1. Your reflection

Today, in the role of managers, we realised that the mathematical formula for conditional probability is used for:

Your answer: 0 characters

2. Where else is conditional probability used?

Choose ONE area outside school and cars where you think similar risk cards are filled in. All three options are real examples – just choose one and justify it.

A) In medicine – to calculate the risk of a disease for people with a particular gene. B) In banking – to assess the risk that a client will not repay a loan if they have changed their job. C) In manufacturing – to estimate whether a product is faulty if a quality check has flagged it.

Summary of the calculations (teaching data)

22% P(crash | violation) = 66/300

7% P(crash | no violation) = 49/700

≈3.14× higher risk for drivers with violations

≈57.4% P(violation | crash) = 66/115 – a different denominator!

18% P(crash | aged 18–24) = 45/250

30% · 6% · 12% profile risk of A · B · C

What does road risk cost?

Facing realistic cases

Client A

Client B

Client C

Task 1. Your first choice

Explore the teaching database

Task 2. Choose between two statements

Calculating the risk mathematically

Task 3. P(crash | violation)

Task 4. P(crash | no violation)

Task 5. How many times higher is the risk?

Task 6. Reversing the condition: P(violation | crash)

Comparison: the same data, a reversed condition

Task 7. Why are the results different?

Task 8. Interpretation: does the number prove a cause?

Age, violations and the client’s profile

Task 9. P(crash | aged 18–24)

Task 10. Profile risk of the three clients

A Client A (age 19, 2 violations)

B Client B (age 36, no violations)

C Client C (age 58, 1 violation)

Task 11. What does age show?

Task 12. The phone at the wheel

Final product: the insurance card

Client A

Client B

Client C

Compare the price ratios and the risk ratios

Task 13. Business argument

Exit card

1. Your reflection

2. Where else is conditional probability used?

Summary of the calculations (teaching data)

How to run the lesson

Before the lesson

During the lesson

For the final part

Practical tips