Six murders in one day in Philadelphia.

There were six homicides in Philadelphia yesterday. The headline in the Philadelphia Inquirer is "Summer's beginning: Six dead in one day". The events happened as follows:

• a triple homicide in North Philadelphia;


• a triple shooting in Kensington -- two died, one was critically wounded;


• one man shot to death in Kingsessing.
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I saw the headline while walking past a newspaper box well before I read the article. I thought "hmm, six murders in one day, is that a lot?" Last year Philadelphia had 406 murders; this year there have been 195 so far, as compared to 177 up until this time last year. The number I carry around in my head is that Philadelphia has one murder a day, although the actual 2006 figure was about 1.11 murders per day.

Since I didn't know that there had only been three incidents, I assumed that the six murders had all been separate. Furthermore, I assumed that murders are committed independently, since the murderers aren't aware of each other's actions. This second assumption seems believable to me. I've heard that, say, school shootings inspire copycats, mostly because they create a media circus around them -- at the time of the Virginia Tech massacres I remember people saying that the media shouldn't cover the shootings so much because they might "give people ideas", and I vaguely recall similar sentiments around the time of Columbine. But a single murder, in a city where the average day sees one murder, doesn't draw much attention.

If the murders are independent, then I figure I can model the random variable "number of murders per day" with a Poisson distribution. The rate of the distribution would be the average number of murders per day, which is 1.11; thus the probability of having n murders in a day should be e^-1.11 (1.11)ⁿ/n!. This leads to the numbers:

n:	0	1	2	3	4	5	6	7+
Prob. of n murders in one day	0.3296	0.3658	0.2030	0.0751	0.0209	0.0046	0.00086	0.00016

So six or more murders should happen in a day about one day in a thousand, or once in almost three years. That seems like an argument for newsworthiness. But on the other hand, let's say there's some lesser crime -- crime X -- that is committed in Philadelphia with such frequency that crime X does not occur on only one day in a thousand. (Such a crime would be something that happens 2516 times per year, or 6.9 times a day.) I don't see that being front-page news. Lots of one-in-a-thousand things happen every day.

Of course, what actually occurred yesterday was not six independent murders. It sounds like there were only three murderers. So it's time for new assumptions. Let's now assume that all murderers act independently, but that two in five of them kills one person; two in five kill two people; one in five kill three people. This means the average murderer kills 1.8 people. Further, let's say that murderers go out and kill people as a Poisson process with rate 0.62 -- that's the old rate divided by 1.8, so there are still the same number of murders.

(The assumptions of how many people a murderer murders are made up, I admit, but the only list of murders I can find are the Inquirer's interactive maps, and it doesn't seem worth the time to harvest the data I'd need from them.)

Now, for example, the probability that three people are murdered on any given day is the sum of the probability that there's one triple homicide, one double and one single, or three single. Running through the computation, I get:

n:	0	1	2	3	4	5	6	7+
Prob. of n murders in one day	0.5379	0.1334	0.1499	0.1012	0.0372	0.0230	0.0103	0.0071

The probability of one or two murders in a day goes down; the probability of zero, or of three or more, goes up. Suddenly yesterday isn't nearly as rare. Days with six or more murders are, under these assumptions, 1.74% of all days -- just over six per year.

The calculation I'm afraid to do -- if I even could do it -- is "how likely am I to get murdered each time I go outside?" Fortunately I live in a decent neighborhood; but some neighborhoods not that far away from me have had some of the worst violence. But it occurred to me that at 400 murders a year, if you live in Philadelphia for 75 years there will be thirty thousand murders in that time span. Philly has about 1.5 million people. So if things stay like they are, the average Philadelphian has a one in fifty chance of dying by murder. In comparison, the nationwide murder rate in 2005 was 5.6 per 100,000; multiplying by an average lifespan of 75 years we get 420 murders per 100,000 people. So one in every two hundred and forty Americans will die of murder, if things stay like they are.