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### 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

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
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/
```
Associated Topics:
High School Probability

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