Base rates
Why rare things stay rare, even after a warning sign.
In your head, jot if needed · no calculator why?
Opens at level 28. You're level 1. You can read and practice here now.
the idea
A base rate is how common something is before any evidence, like 2% of accounts being fraudulent. When the thing is rare, even a good filter raises many false alarms, because the honest group it can be wrong about is so much bigger.
Two more habits belong here. Over many independent tries, the chance of something happening at least once grows fast, but not by adding the chances. And an extreme result is partly luck, so the next one usually lands closer to average: regression to the mean.
techniques
Count a crowd
- Split the population into counts: the rare group and the rest.
- Flag each group at its own rate: true flags and false flags.
- The chance a flag is real is true flags ÷ all flags.
worked example
Out of 1,000 accounts, 5% are fraudulent. A filter flags 80% of fraudulent accounts and wrongly flags 10% of honest ones. What percent of flagged accounts are fraudulent? Round to the nearest percent.
- Fraud: 50 accounts, 40 of them flagged.
- Honest: 950 accounts, 95 of them flagged.
- 40 ÷ 135 ≈ 29.6%, so about 30%.
Answer: 30%
One minus none
- Chance it does not happen on one try: 1 minus the chance.
- Multiply that by itself once per try: the chance of none at all.
- At least once is 1 minus the chance of none.
worked example
Each shipment has a 10% chance of arriving damaged. Over 3 independent shipments, what is the chance that at least one arrives damaged? Round to the nearest percent.
- No damage on one: 0.9.
- None in 3: 0.9 × 0.9 × 0.9 = 0.729.
- At least one: 1 − 0.729 = 0.271, about 27%.
Answer: 27%
watch out for
- Reading "flags 90% of fraud" as "90% of flags are fraud". The second depends on how rare fraud is.
- Adding the chances for "at least once": 5% ten times is not 50%. Multiply the chances of none, then subtract from 1.
- Crediting a change for the rebound after an extreme result. The luck part rarely repeats, so some drift back toward average comes anyway.
practice
What a flag means
worked example
Out of 10,000 parts, 5% are defective. An inspection machine flags 90% of defective parts and wrongly flags 2% of good ones. What percent of flagged parts are actually defective? Round to the nearest percent.
Answer: 70%
- 500 defective → 450 flagged (90%). 9,500 good → 190 flagged (2%).
- 450 of the 640 flags are real: 450 ÷ 640 ≈ 70.3%, about 70%.
At least once
worked example
Each product launch has a 10% chance of a serious bug. Over 18 independent launches, what is the chance of at least one serious bug? Round to the nearest percent.
Answer: 85%
- The chance of none in all 18 is 0.9¹⁸ ≈ 0.15.
- So at least one: 1 − 0.15 = 0.85, about 85%.
Regression to the mean
worked example
The restaurant with the worst inspection score in town is inspected again the next month. Assuming nothing changed in the kitchen, what should you expect from its second score?
- an even more extreme result
- exactly average results
- no change
- some improvement, since extreme results tend to drift back toward average
Answer: some improvement, since extreme results tend to drift back toward average
- An extreme result is usually partly real and partly chance. The chance part tends not to repeat, so the next result usually lands closer to average.
- That's regression to the mean: expect some improvement even when nothing changes.