Percentages are compact but slippery
A percentage squeezes a lot into a small space. “A test is 80% sensitive with a 10% false positive rate, and the condition affects 1% of people” is four numbers, each a share of a different group. Working out what a positive result means from that sentence is hard, even for people who are comfortable with numbers.
Natural frequencies replace those percentages with counts of people, cases or events in a fixed, round-sized group. The idea is the same one used when explaining Bayes theorem with a tree, but stripped down to its simplest form: pick a group of 1,000, count what happens to each part of it, and read off the answer.
The conversion, step by step
The figures below are illustrative only, not data about any real test.
Start with: prevalence 1%, sensitivity 80%, false positive rate 10%.
Step 1. Choose a group. Take 1,000 people.
Step 2. Split by the starting rate. 1% of 1,000 is 10 people with the condition, so 990 without it.
Step 3. Apply the test to each group. 80% of the 10 is 8 who test positive, leaving 2 missed. 10% of the 990 is 99 who test positive, leaving 891 true negatives.
Step 4. Keep only the positives. 8 + 99 = 107 people test positive.
Step 5. Answer the question. Of 107 positives, 8 have the condition. That is 8/107, about 7.5%.
Every number in the chain is a whole number of people. There is no step where you multiply one percentage by another and wonder what the result refers to. If you do the same problem with the formula, you get 0.8 × 0.01 / (0.8 × 0.01 + 0.1 × 0.99) = 0.008 / 0.107, which is again about 7.5%. The counts and the formula agree, but the counts show why.
Why counts help
There are a few plausible reasons the counting version feels easier, though it is wise to say “often” rather than “always”.
- Everything shares a reference group. Percentages apply to different groups, such as 80% of those with the condition and 10% of those without. Counts out of 1,000 put everything on the same footing.
- The base rate is built in. When you split 1,000 into 10 and 990, the rarity of the condition is visible, so it is hard to ignore. This is the fix for base rate neglect.
- Small numbers are easier to picture. “8 out of 107” is easier to hold in mind than 0.0748.
- The arithmetic is simpler. You mostly multiply and add whole numbers rather than dividing one probability by another.
Dealing with awkward numbers
Sometimes 1,000 does not divide cleanly. If the false positive rate were 9%, then 9% of 990 would be 89.1 people. You have two choices: use a bigger group, such as 10,000, so the counts come out whole, or accept a decimal and treat it as an average. Either is fine, as long as you keep to one group size throughout. Rounding early is the more likely source of error than using fractions.
It also helps to check that the groups add up. In the example above, 8 + 2 + 99 + 891 = 1,000. If the pieces do not sum to the starting total, something has gone wrong.
Risk changes: absolute versus relative
Natural frequencies are just as useful when reading claims about a change in risk. Suppose a headline says something “doubles your risk”. That could mean going from 1 in 1,000 to 2 in 1,000, or from 100 in 1,000 to 200 in 1,000. The relative change, a doubling, is identical, but the absolute change is 1 extra person per 1,000 in the first case and 100 extra per 1,000 in the second.
Putting both in counts out of 1,000 makes the real size of the change visible. It also helps with the difference between percentage points and percentages: going from 0.1% to 0.2% is a rise of 0.1 percentage points, even though it is a 100% increase in relative terms.
Chance in games, restated as counts
The same trick clears up game odds. A chance of 1/6 on a die is about 167 outcomes out of 1,000 rolls on average, not a promise about any set of rolls. Saying “roughly 167 out of 1,000” rather than “16.7%” also reminds you that actual results will vary around that figure. The more rolls you make, the closer the share tends to be to 1/6, which is the idea behind the law of large numbers, though the exact count in any one run is uncertain.
A word of caution for games of chance: thinking in counts clarifies what the numbers mean, but it cannot remove a house edge. Each play remains independent, past results do not change future odds, and gambling can be harmful. If it is becoming a problem, a local support service can help.
When to use the method
Reach for natural frequencies whenever a problem gives you several percentages that apply to different groups, or when you have a conditional question such as “given this clue, how likely is that?” Pick a round group size, split it step by step, drop the cases that do not match the clue, and compare what remains. It will not do anything a formula cannot, but it makes mistakes much easier to spot.