Coaching Centre
Probability in health and insurance
Using probability and expected frequency to make real decisions: blood types and donor matching, reading medical test results with an expected-frequency table, and how insurers use claim rates to set premiums.
Probability isn't only about dice and raffles. Hospitals plan blood supplies with it. Doctors use it to explain what a test result really means. Insurers use it to decide what to charge you for car or contents insurance. The maths is the same as the last two sections: a probability from data, multiplied by a number of people. The skill is reading the situation carefully.
Blood types and donors
Every person's blood has a type made of two parts. The ABO group (O, A, B or AB) depends on which markers, called antigens, sit on the red cells: A, B, both, or neither. The Rh factor is positive (+) if the cells carry the RhD marker and negative (−) if not. A patient can safely receive red cells only if the donor's cells carry no marker the patient doesn't already have.
The table shows approximate percentages of each type among Australians, based on figures published by Australian Red Cross Lifeblood. They are rounded and vary a little between sources.
| Type | O+ | O− | A+ | A− | B+ | B− | AB+ | AB− |
|---|---|---|---|---|---|---|---|---|
| Approx. % | 40 | 9 | 31 | 7 | 8 | 2 | 2 | 1 |
A Lifeblood donor centre expects 2500 donors this month. (a) How many would you expect to be O−? (b) A patient has type A+. What is the probability that a randomly chosen donor can give red cells to this patient?
Choose the patient's blood type. The donor types that can safely give red cells light up green, and the readouts give the probability and the expected number of matching donors.
Which patient type has the fewest matching donors? Which has the most? Set the slider to 300 donors and find how many would match a B+ patient.
What does a positive test really mean?
Medical tests aren't perfect. A test might correctly pick up $95\%$ of people who have a condition, but also wrongly flag $5\%$ of people who don't. When a condition is rare, those false alarms can outnumber the real cases. The clearest way to see this is to imagine a round number of people and fill in an expected-frequency table.
A condition affects $2\%$ of people. A screening test gives a positive result for $95\%$ of people who have it and for $5\%$ of people who don't. Imagine 1000 people are tested. Find the probability that a person with a positive result actually has the condition.
| Test positive | Test negative | Total | |
|---|---|---|---|
| Has condition | 19 | 1 | 20 |
| No condition | 49 | 931 | 980 |
| Total | 68 | 932 | 1000 |
So only about 28 in every 100 positive results are real. This is why a positive screening test is usually followed by a second, more accurate test before anyone is diagnosed.
How insurers set premiums
An insurer can't know which customers will make a claim, but from years of data it knows the claim rate. It uses expected frequency to estimate the total it will pay out, then shares that cost across all customers.
Records show 3 claims per 1000 home contents policies each year, with an average claim of $\$15{,}000$. An insurer has 20{,}000 policies. Find the expected number of claims, the expected total payout, and the break-even premium per policy.
Your turn
A car insurer finds that $4\%$ of its policyholders make a claim each year, and the average claim is $\$6{,}500$. What is the break-even premium per policy?
Show the answer
$\text{break-even premium} = 0.04 \times \$6{,}500 = \$260$. Out of every 100 policies, 4 claims cost $4 \times \$6{,}500 = \$26{,}000$, and $\$26{,}000 \div 100 = \$260$.
About $40\%$ of Australians have type O+ blood. How many of 2500 donors would you expect to be O+?
$2500 \times 0.40 = 1000$.
About $1\%$ of Australians are AB−. How many of 400 donors would you expect to be AB−?
$400 \times 0.01 = 4$.
A patient has type B−. Which donors can give them red cells?
A B− patient has the B marker but not RhD. Donor cells must carry no A and no RhD, so only O− and B− are safe.
Using the table on this page, $59\%$ of donors can give red cells to a B+ patient. How many of 300 donors would you expect to be suitable?
Compatible donors for B+ are O+, O−, B+ and B−: $40 + 9 + 8 + 2 = 59\%$. Expected number $= 300 \times 0.59 = 177$.
Which patient type can receive red cells from the largest group of donors?
AB+ patients already carry A, B and RhD, so any donor's cells are safe. That is $100\%$ of donors.
An insurer records 3 claims for every 1000 policies each year. How many claims should it expect from 20{,}000 policies?
$P(\text{claim}) = \frac{3}{1000} = 0.003$. Expected claims $= 20{,}000 \times 0.003 = 60$.
A car insurer finds $4\%$ of policies lead to a claim each year, with an average claim of $\$6{,}500$. Find the break-even premium in dollars.
Break-even premium $= 0.04 \times \$6{,}500 = \$260$.
Why does an insurer charge more than the break-even premium?
The break-even premium only covers the expected claims. Staff, offices, advertising and profit all need extra money.
Exam style: A condition affects $2\%$ of people. A test gives a positive result for $5\%$ of people who do not have the condition. Out of 1000 people tested, how many would you expect to be false positives?
People without the condition: $1000 \times 0.98 = 980$. False positives: $980 \times 0.05 = 49$.
Exam style: A condition affects $1\%$ of 10{,}000 people. A test is positive for $90\%$ of those with the condition and $8\%$ of those without it. Find, as a decimal to one decimal place, the probability that a person with a positive result has the condition.
With condition: $10{,}000 \times 0.01 = 100$, and $100 \times 0.9 = 90$ test positive. Without: $9900$, and $9900 \times 0.08 = 792$ test positive. $P = \frac{90}{90 + 792} = \frac{90}{882} \approx 0.102$, which is $0.1$ to one decimal place.