Murder Rate Probability

Date: 01/12/98 at 10:13:02
From: J. R. Lankford
Subject: Help: murder rate probability problem

I found your wonderful Dr. Math site in a desperate search of the 
Internet when I ran into a probability problem for the murder mystery 
I'm writing. The question is: if a Midwestern town has a murder rate 
of 11 murders per year, what are the chances that 3 of them will occur 
on the same day?

I used your "Probability of Being Born on Monday" example as a model 
and came up with the following formula:

   [1/365*(1-1/365)^11]*3

   Answer = .0079968

Is that correct?  And is that very rare?  I'd hate for math whizzes to
laugh at my detective's calculations and conclusions.

Thanks,
Mrs. JRL
Administrator
WriteLab  http://www3.hypercon.com/~jilla   
NovelDoc  http://www3.hypercon.ocm/~jilla/NovelDoc   

Date: 01/12/98 at 16:34:33
From: Doctor Anthony
Subject: Re: Help: murder rate probability problem

This is actually a Poisson distribution problem.  The average murders 
in 365 days will be 11.

So in 1 day the average is 11/365

P(no murders on a particular day) = e^(-11/365) =  0.9703  =  P(0)

P(1 murder on a particular day) = (11/365) x P(0) = 0.02924

P(2 murders on a particular day) = (1/2)(11/365) x P(1) = 0.0004406

P(3 murders on a particular day) = (1/3)(11/365) x P(2) = 0.000004426

Now if we consider 3 murders in a day a 'success' in terms of 
binomial probability, the chance of at least 1 success in 365 trials 
(a complete year) is 

            1 - Prob(no successes)

            1 - .999995573^365

            1 - .99838563

            0.001614

So the probability of at least one day in a year producing 3 murders 
is 0.001614,  which is certainly very rare.  

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 01/12/98 at 17:47:40
From: J. R. Lankford
Subject: Re: Help: murder rate probability problem

Thank you so much for that response.  It's been so many years since I 
had a math book in my hands, I never would have come up with it. What 
a great service.

Mrs JRL

Date: 03/18/98 at 10:31:27
From: J. R. Lankford
Subject: Help: the plot thickens

In January you very kindly helped me with the following question for 
the murder mystery I'm writing:

	If a midwestern town has a murder rate of 11 murders per year,
	what are the chances that 3 of them will occur on the same day?

You gave me a Poisson distribution. From it, my detective deduced that 
the town could expect a day with 3 unrelated murders about once in 620 
years.

Now the plot has thickened, as plots tend to do. At the start, my
detective now doesn't know all three deaths are murders, much less 
that they're by the same person. Instead, he must deduce this 
possibility.

What are the chances of the following unrelated events occurring on 
the same day in the same midwestern town of 185,000?

	1 murder            murder rate = 11/year
	1 accidental death	non-vehicular accidental death rate = 33/year
	1 police shooting	rate = .264/year *


   *assuming a national rate of 383/year for a population of 260  
    million, and that police shootings are 3% less likely to occur in 
    the Midwest than average.

Do I use your Poisson distribution model to calculate each probability
separately, then multiply them by each other? Any help you can give 
will be much appreciated and gratefully acknowledged in the novel when 
(cross fingers) it is published.

Thanks,
Mrs. JRL

Date: 03/19/98 at 16:05:34
From: Doctor Anthony
Subject: Re: Help: the plot thickens

On a particular day, probability of 1 or more murders is  1 - P(0)

   P(0) = e^(-11/365) =  0.9703126

                     Probability 1 or more murders = 0.029687

On same day, probability of 1 or more accidental deaths is 1 - P(0)

   P(0) = e^(-33/365) =  0.913555

                     Probability 1 or more accidents = 0.086444

On same day probability 1 or more police shootings is 1 - P(0)

   P(0) = e^(-.264/365) =  0.999277

                     Probability 1 more police shootings = 0.000723

If all three happen on the same day we multiply the probabilities.

       0.00000185547

The chance that on a day this does not happen is 0.999998144

In 365 trials the probability that it happens 1 or more times is:

     1 - .999998144^365

   =  0.000677

This is a lower probability than the three murders. The very small 
probability of the police shooting has reduced the combined 
probability. 

-Doctor Anthony,  The Math Forum
 http://mathforum.org/dr.math/