Murder Rate ProbabilityDate: 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/ |
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