Is a 99% accurate test right 99% of the time? Not when the thing it looks for is rare. Take a condition that 1% of the tested group has, and a test that catches 99% of real cases and clears 99% of people who do not have it. A positive result from that test is right exactly 50.00% of the time. Half. The other half are false alarms, and none of them comes from a faulty test.
That answer surprises almost everyone the first time, and the reason is not medical at all. It is conditional probability: the chance of a positive among the sick is not the chance of being sick among the positives. The rest of this article takes the number apart, then changes one assumption at a time to see what moves it and what does not.
Where does the other half come from?
Numbers are easier than percentages here, so picture 10,000 people walking into a screening. At 1% prevalence, 100 of them have the condition and 9,900 do not.
The test finds 99 of the 100. It misses one. So far so good.
Now the healthy side. A specificity of 99% means 1% of healthy people still test positive, and 1% of 9,900 is 99. The pile of positive results holds 99 people who are ill and 99 who are not: 198 positives, of which 99 / 198 are real. That is the 50.00%, and the table below is what the calculator prints for these inputs.
| Out of 10,000 tested | Positive | Negative | Total |
|---|---|---|---|
| Have the condition | 99 | 1 | 100 |
| Do not | 99 | 9,801 | 9,900 |
| Total | 198 | 9,802 | 10,000 |
Written as Bayes' theorem it is the same sum: P(sick | positive) = 0.01 × 0.99 / (0.01 × 0.99 + 0.99 × 0.01). The two products are equal, so the fraction is exactly one half. Nothing about this depends on the disease, the lab or the country. A spam filter, a fraud rule or a metal detector at an airport runs on the same arithmetic.
The negative side of the same screening looks very different. Of 9,802 negatives, only one person is ill, so a negative result is right 99.99% of the time. A rare condition makes a negative almost certain and a positive a coin toss, which is close to the reverse of what the word accurate suggests.
What if the condition is more or less common?
Prevalence is the number that moves everything, and it is the one that never appears on a test's label. The next table keeps the test fixed at 99% sensitivity and 99% specificity and changes only how common the condition is in the tested group. The middle column counts positives per 10,000 people; the last one is explained two sections further down.
| Prevalence | Positives per 10,000 | Chance a positive is right | After a second positive |
|---|---|---|---|
| 0.1% | 110 | 9.02% | 90.75% |
| 0.5% | 149 | 33.22% | 98.01% |
| 1% | 198 | 50.00% | 99.00% |
| 2% | 296 | 66.89% | 99.50% |
| 5% | 590 | 83.90% | 99.81% |
| 10% | 1,080 | 91.67% | 99.91% |
| 20% | 2,060 | 96.12% | 99.96% |
At 0.1%, the kind of rate seen when a whole population is screened for something uncommon, a positive is right only 9.02% of the time. At 10%, which is closer to a clinic seeing people who already have symptoms, the same test is right 91.67% of the time. Same test, same lab. The only difference is who walked in.
This is why the question "how accurate is the test?" has no single answer. A test does not have a positive predictive value. A test used on a particular group does.
What if a few details of the test change?
What if the specificity is 95% instead of 99%?
Four percentage points sound small. At 1% prevalence they are not: a test with 99% sensitivity and 95% specificity has a positive predictive value of 16.67%, one in six. The healthy group is 99 times larger than the sick one, so every point of specificity is multiplied by that ratio. Improving sensitivity from 99% to 100% would add at most one true positive per 10,000; improving specificity from 95% to 99% removes 396 false ones.
What if you test again?
Here the arithmetic turns friendly. A second positive on an independent test raises the chance from 50.00% to 99.00%. The quickest way to see why is the likelihood ratio, which the calculator prints as LR+. For 99% and 99% it is 0.99 / 0.01 = 99. Before testing, the odds of being ill were 1 to 99. One positive multiplies them by 99, giving 1 to 1, the coin toss. A second positive multiplies them by 99 again: 99 to 1, which is 99%.
The catch sits in the word independent. Running the same sample through the same machine twice repeats the same mistakes, because whatever fooled it the first time is still in the tube. The 99% applies to a different kind of test, or at least a fresh sample, and the calculator states that assumption under its result.
What if ten healthy people are tested?
Now turn the question around. Ten people, none of them ill, each with a 1% chance of a false positive. How likely is it that the batch comes back clean? Exactly 90.44%, which is 0.99 to the tenth power. Put the other way round, the chance of at least one false alarm is 9.56%, so roughly one batch in ten comes back with a healthy person flagged. With 100 healthy people it rises to 63.40%, and a false positive somewhere in the group becomes more likely than not.
Exactly one false positive among ten has probability 10 × 0.01 × 0.999 = 9.14%. Exactly two: 45 × 0.012 × 0.998 = 0.42%. Where do the 10 and the 45 come from? They count the ways to pick which people get the false result, and they are the coefficients of the binomial expansion of (x + 1)10, row 10 of Pascal's triangle: 1, 10, 45, 120, 210, 252 and back down.
The sum of that row, 210 = 1,024, is the number of possible positive and negative patterns for ten people. Most of them are wildly unlikely at a 1% error rate, which is why the probabilities fall so fast after the first two terms: three false positives among ten healthy people has a probability of about 0.01%.
What if the retest is booked every 45 days?
A different kind of question, but one that comes up whenever a screening is repeated on a fixed schedule: on which weekday does each retest land? Count the days, divide by 7 and keep the remainder. 45 mod 7 = 3, because 45 = 7 × 6 + 3, so a test first done on a Monday comes back on a Thursday.
Keep going and the remainders run 3, 6, 2, 5, 1, 4, 0: Thursday, Sunday, Wednesday, Saturday, Tuesday, Friday, and on the seventh retest, after 315 days, Monday again. Every weekday is visited once before the cycle closes, because 45 and 7 share no common factor. A 42-day interval would hit Monday every time, since 42 is on the list of numbers divisible by 7. For a clinic that is closed on weekends, that difference matters more than the arithmetic suggests.
The same remainder cycles answer puzzles that look nothing like a calendar: the last digit of 7100 is 1 because last digits of powers of 7 repeat every four steps, and the remainder of 2100 divided by 7 is 2 because powers of 2 cycle through 2, 4, 1.
All the scenarios side by side
Each row changes one thing against the starting point of a 1% condition and a 99% / 99% test. The last column is the honest one-line verdict.
| Scenario | Chance a positive is right | Is the positive worth acting on? |
|---|---|---|
| One test, 1% prevalence | 50.00% | not on its own; confirm first |
| One test, 0.1% prevalence (mass screening) | 9.02% | no; most positives are false |
| One test, 10% prevalence (people with symptoms) | 91.67% | usually yes |
| One test, 1%, specificity 95% | 16.67% | no; five false alarms per real case |
| Two independent positives, 1% | 99.00% | yes; this is what "99%" was supposed to mean |
| Ten healthy people, 1% false positive rate | 9.56% chance of at least one alarm | expect one false alarm in about every tenth batch |
So, is it right 99% of the time?
It is right 99% of the time about the people who have the condition, and 99% of the time about the people who do not. It is not right 99% of the time about a positive result, because that number depends on how many people in the tested group had the condition in the first place, and the label never says. Ask for the prevalence before you trust a positive, and ask for a second, independent test before you act on one. The arithmetic takes ten seconds with the calculators below, and it is the same arithmetic whether the test is for an illness, a fraudulent card payment or a bag at airport security.
Tools discussed in this article
- Conditional probability calculator: P(A | B) from a two-way table or from P(A and B) / P(B), Bayes' theorem from P(B | A), total probability over two or three hypotheses, and a diagnostic test mode with PPV, NPV, LR+, LR- and the effect of a second positive, all in exact fractions.
- Binomial expansion calculator: (2x - 3)4, (x + 1/x)6 or (a + b)n written out term by term, the coefficient of any power of x, exact C(n, k) up to n = 1000 and Pascal's triangle to row 30.
- Modulo calculator: a mod b for negative numbers, decimals and integers of any length in the Python, JavaScript and Euclidean conventions, powers mod n with their cycle, tables of remainders and divisibility checks.
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