What was the probability that Zimmerman was right?

7/16/2013 05:15:00 PM
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Update: I have updated this post with better data here.

Matt Yglesias has a post pointing out that Richard Cohen continues to argue the "Prosecutor's Fallacy," this time arguing that George Zimmerman was right to suspect Trayvon Martin of a crime because black people are more likely to commit crimes. The error in that logic, as Yglesias points out, has to do with Bayes' Theorem. I'm a big fan of Bayes Theorem.

Lets define two events. Event $A$ is that the person is black. Event $B$ is that the person is a violent criminal. I don't have data on the racial makeup of crimes, but lets assume that NYC police commissioner Ray Kelly is correct that 75 percent of crimes are committed by black people. Thus, we have that $$Pr \left(A|B\right)=0.75$$ But what Cohen wants to know is actually the reverse, the probability of being a criminal given that the person is black: $$Pr \left(B|A\right)$$ Bayes' Rule gives us a way to find the latter using the former: $$Pr \left(B|A\right)=\frac{Pr\left(B\right)Pr \left(A|B\right)}{Pr\left(A\right)}$$ But, to do this, it turns out we need extra information, in particular we need $Pr\left(A\right)$ and $Pr\left(B\right)$. That was Matt Yglesias' whole point.

I don't know $Pr\left(B\right)$. Neither does Cohen, which is why he should shut up. But just for kicks, lets use the national violent crime rate as a proxy. This almost certainly overstates the true value, since there are lots of repeat offenders out there, but lets roll with it. Thus using 2011 data we have $$Pr\left(B\right)=\frac{386.3}{100,000}$$ And we will use the percent of the population that is black to get $$Pr\left(A\right)=0.136$$ Plugging into Bayes' Theorem gives us $$Pr \left(B|A\right)=\frac{0.003863\left(0.75\right)}{0.136}=0.0213$$

So in reality, Zimmerman shot Trayvon to death over a 2.13 percent chance that he would commit some type of violent crime. (Update, a commenter pointed out I accidentally inserted an extra zero in the calculation. This has been corrected.)

The lesson here is quite unambiguous: race is not a meaningful indicator of criminality. I shouldn't have to tell you people this.
Anonymous 7/16/2013 06:49:00 PM
You're off by a factor of 10 (although your point still holds). 386.3/100000=0.003863, not 0.0003863. So, the % chance is 2%, not 0.2%.
Matthew Martin 7/16/2013 07:42:00 PM
Corrected. Thanks!
Unknown 7/16/2013 07:39:00 PM
No Zimmerman shot Martin because he was getting pummeled and feared for his life. In order for your study to be valid, you'd need the relevant variables for the community where this incident took place since crime is local.

Zimmerman was concerned with property not violent crime in particular burglaries. Reuters wrote a pretty good article about this case last year here http://tinyurl.com/c6evjrp
In it we learn 8 burglaries occurred within a 14 month span before the shooting but let's just use the burglary rate for Sanford County as a whole which in 2011 was 1741 per 100000 residents which is more than double the national rate BTW. http://www.city-data.com/city/Sanford-Florida.html

We don't know how many of the burglary suspects in Sanford are Black although we do know nearly all of the recent burglary suspects in Zimmerman's development were Black.

Let's just use your 75% violent NYC number as a placeholder for now which is probably very conservative in relation to this actual development. Now I think we are in business.

Pr A=.30 Blacks make up 30% of Sanford and it's close to their percentage of the actual development.
Pr B= .01741
Pr (A/B)=.75

Plugging these variable into the theorem I come up with 4.35% chance that Zimmerman comes across a Black burglary suspect. Of course we know elderly Black women aren't burglarizing homes so the Black proportion number is over stated. I suspect the actual probability that Zimmerman came across a random young Black male that is a burglary suspect is in excess of 20% if using applicable numbers. A reasonable proxy for Zimmerman to act.
Anonymous 7/16/2013 10:08:00 PM
so 20% is a reasonable chance of justifiably initiating a deadly confrontation with a teenager to you.

you wouldn't bet $100 on a 20% chance of doubling your money. that's worse than three card monte odds. but a human life is all gravy i guess.

your wording is pretty slimy too. 20% chance of coming across a "burglary suspect." not even "a burglar." a "burglary suspect."
Anonymous 7/16/2013 10:11:00 PM
maybe cut out the middle man and just put one in every five black dudes on death row so we don't have to waste your beautiful mind on trivial matters such as murder.

oh shit i think we already do that.
C. E. W: III 8/15/2013 09:38:00 PM

"maybe cut out the middle man and just put one in every five black dudes on death row"

No, preferably in five out of five everywhere.