Learning to Think - The Eyewitness

If you want to be a good thinker you have to stretch your brain with hard problems, problems that are over your head. This problem might be over my head but I THINK I understand it:

Suppose you live in a city divided between two racial or cultural groups - we'll call them A's and B's. In this city 15% of the people are A's and 85% are B's.

There has been a crime and we have an eyewitness who says the perpetrator was an A. The police test this witness and conclude that he correctly identifies A's and B's 80% of the time.

The police chief wants to use his resources wisely so he orders his detectives to focus first on suspects who are A's....

Given that our eyewitness says the criminal is an A and the eyewitness correctly identifies A's vs. B's 80% of the time this must be a good decision, right? Or is it?

Answer below.

*

*

*

*

*

*

*

*

This is an interesting problem and one that will repay careful study. One of the most important lessons we learn from this problem is that a seemingly obvious correct answer can, in fact, be wrong.

If all we know is that an eyewitness says that an A committed the crime rather than a B, and that this witness can distinguish between A's and B's correctly 80% of the time, it seems obvious that the crime was most likely committed by an A.

But in this case we know something else: we know that A's make up only 15% of the population while B's make up the other 85%. The problem doesn't say whether A's or B's are more likely to commit crimes so, for this problem, we should assume there is no significant difference in criminal tendencies between the two groups.

***

[Check out this great book about thinking better.]
[If you want to support "Anything Smart" just click on book links like the one below to buy your books. "Anything Smart" will receive a commission. Thanks!]

***

So let's do a thought experiment: let's imagine our witness saw 100 crimes rather than just one, and let's imagine that 85 of those crimes were committed by B's and 15 were committed by A's since that is the proportion we would expect to find given no difference in propensity to crime between the two groups.

If our witness saw 85 crimes committed by B's he will correctly tell us the criminal was a B 68 times but he will INCORRECTLY tell us the criminal was an A 17 times because he correctly distinguishes between A's and B's only 80% of the time.

On the other hand, if our witness saw 15 crimes committed by A's he will correctly tell us the criminal was an A 12 times and INCORRECTLY tell us the criminal was a B 3 times.

Looking at these two scenarios we notice that our witness will tell us, after seeing 100 crimes committed, that an A was the criminal 29 times even though, in reality, an A was the criminal only 15 times.

More importantly, out of 29 times our witness tells us that an A committed the crime he will be RIGHT only 12 times (41%) and WRONG 17 times (59%.)

In conclusion even though our witness has an 80% success rate of distinguishing between A's and B's and even though he identifies a particular criminal as an A it is still more likely, given the facts in this problem that the criminal was a B.

Without the eyewitness we would have have evaluated the probability of an A being the criminal at 15%. With the eyewitness the probability of an A being the criminal goes up to 41% but that is still below 50% so it would be a terrible mistake for the police chief to focus his resources on suspects who are A's. In the absence of any other evidence we have to conclude that our criminal is most likely a B.

***

When we don't understand something we tend to think about it and try to figure out.

When we ASSUME we know something we stop thinking about it, and this can result, sometimes, in tragic consequences.

It would be wise to think about things we "know" just as much as we think about things we don't know, maybe even more!

Copyright © 2017 by Joseph Wayne Gadway