We discussed the Prosecutor’s Fallacy in the past, but I recently found an excellent example of it in our local paper. To recap, the classic example of the fallacy goes something like this:

In the maneuvering for what data was to be allowed in OJ’s murder trial, the prosecutor wanted to introduce the fact that OJ had previously battered Nicole. The defense moved to prevent the jury from being exposed to this information because it would be prejudicial to their client and it was not relevant. The prosecutor wants to present anything that could lead a jury to think the defendant is capable of uncontrolled rage, but the defense introduces statistics that show only 1 out of 2,500 men who batter a woman later go on to kill her. This number is so small that it shows nothing can be gained by exposing a jury to the history, which is irrelevant by these numbers. (BTW, death from a person who previously battered is also relatively rare. The current death rate in her demography is 45/100,000.)

So what would you do if you were the judge? Think about it before reading on, or if you must read on, remember what you read and try to stump your friends with it.

The simple analysis is to consider a pool of 100,000 battered women. By the above, we know that 45 of them will be killed. However, if we divide 100,000 by 2,500, we see that 40 of them will be killed by the person who previously battered them. In other words, 8 out of 9 women killed by batterers were previously battered by them. Does this analysis change your opinion?

Note the defense did not lie. We could be generous and suggest they really believed what they said. After all, the statistics are valid. But they posed the issue incorrectly. By concentrating on batterers, they missed the essential point of considering the history of their victims.

This fallacy bears repeating because it keeps coming up. This week a concerned citizen wrote in our local newspaper to complain about the mayor and City Council of San Diego voting in favor of supporting gay marriage. The first sentence of the editorial is “In a study published by the Archives of Sexual Behavior [sic: no reference], 86 percent of pedophiles described themselves as homosexual.” The editorial goes on to consider the dire effects of allowing marriage and by definition, adoption of children by gay couples and how that would lead to sexual exploitation.

The problem is that I did a quick search using only Google and found several references to child abuse statistics. In general, the largest single block of child molesters are the immediate family with close friends of the family running second. In one study I found, out of nearly 400 children who had been examined in a hospital after being molested, less than 1 percent were assaulted by homosexuals.

If we follow the logic of the editorial to fight child abuse in light of the statistics, instead of concentrating on gays, we are forced to conclude that we should prevent all members of the immediate family from having any contact with children. Depending on how much funding is available for the child protection system, we could also prevent contact with close family friends and neighbors. Obviously this reasoning is flawed.

My quick and dirty survey is not conclusive, but the essential point is there. The editorialist was either misled into the prosecutor’s fallacy or deliberately stated the facts to bias readers toward a particular point of view. Whatever your political preference, let us at least present the data correctly.

In response to the interest my original tutorial generated, I have completely rewritten and expanded it. Check out the tutorial availability through Lockergnome. The new version is over 100 pages long with chapters that alternate between discussion of the theoretical aspects and puzzles just for the fun of it. Puzzle lovers will be glad to know that I included an answers section that includes discussions as to why the answer is correct and how it was obtained. Most of the material has appeared in these columns, but some is new. Most of the discussions are expanded compared to what they were in the original column format.

[tags]decision theory, statistics, odds[/tags]