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Let's work through a few more examples. Imagine that the base rate is only 1%, and the accuracy rate (as described above) is still 99%. Now, there are only 100 drunk people who take a breath test, and 99 of them get arrested. There are 9,900 sober people who take a breath test, and 99 of them get arrested too. There's only a 50-50 chance that your client was one of the drunk ones!
Let's bump the base rate back to 2% again, but now the test is only 95% accurate (the false negative and false positive rates are both 5%). Of the 200 drunk people, 190 get caught. Of the 9800 sober people, 490 get mistakenly arrested(!!!): it turns out that even though our client blew a positive result on the "95% accurate test," there's only about a 28% chance that they were actually drunk. If that's the only evidence the prosecution has, you shouldn't be taking a plea bargain, you should be filing a motion to dismiss!
Statisticians and scientists call the mistake that the prosecutor wanted you to make "the base rate fallacy"--the error of neglecting the rules of "conditional probability" and not taking into account the base rate of some tested-for fact in figuring out how much information we learn from a test. But I prefer to call it the "base rate malpractice trap," because, if you fall into it, you'll give lousy advice to your clients and they really oughta sue you.
There's a simple mathematical formula that you can use to calculate problems like this. Let's use some variables, like in algebra. We'll call the actual chance that your client was drunk (the probability we're trying to calculate): *d*. Let's call the base rate: *b*. Call the false positive rate: *fp*. And call the false negative rate *fn*. Then you can plug the numbers into the following equation:
$$d = \frac{b \cdot (1 - fn)}{(b \cdot (1- fn)) + ((1-b) \cdot fp)}$$
Let's take our last example and apply this formula:
$$d = \frac{0.02 \cdot (1 - 0.05)}{(0.02 \cdot (1- 0.05)) + ((1-0.02) \cdot 0.05)}$$
With some simplification steps:
$$d = \frac{0.02 \cdot 0.95}{(0.02 \cdot 0.95) + (0.98 \cdot 0.05)}$$
$$d = \frac{0.019}{0.019 + 0.049}$$
$$d = \frac{0.019}{0.068}$$
$$d \simeq 0.279$$
which is what we got before, and what our calculator tells us.
### Further reading
- For a detailed and mathier explanation of conditional probability and lots more in basic probability, you can see [a lesson I gave to my students in a data science for lawyers course](https://sociologicalgobbledygook.com/the-basics-of-probability.html).
- For an academic paper reporting on a variety of evidence that judges fall into the base rate fallacy, see Christian Dahlman, Frank Zenker & Farhan Sarwar, [Miss rate neglect in legal evidence](https://academic.oup.com/lpr/article/15/4/239/2580528), 15 Law, Probability and Risk, 239 (2016).
#### Footnotes