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## Probability

by: Otilia Murray I

67

0

1

# Probability MAT 135A

Otilia Murray I
UCD
GPA 3.88

Staff

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## Popular in Mathematics (M)

This 1 page Class Notes was uploaded by Otilia Murray I on Tuesday September 8, 2015. The Class Notes belongs to MAT 135A at University of California - Davis taught by Staff in Fall. Since its upload, it has received 67 views. For similar materials see /class/187423/mat-135a-university-of-california-davis in Mathematics (M) at University of California - Davis.

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Date Created: 09/08/15
Poisson distribution and law Assume a crime has been committed It is known that the perpetrator has certain characteristics which occur with small frequency 1 say 10 8 in the population of size n say 108 A person who matches these characteristics has been found at random eg at a routine traf c stop or by airport security and since 1 is so small charged with the crime There is no other evidence What should the defense be Lets start with a mathematical model of this situation Assume that N is the number of people with given characteristics This is a Binomial random variable but given the assumption we can easily assume its Poisson with A np Choose a random person from among these N call that person 0 the criminal Then independently choose at random another person A who is arrested The question is whether 0 A that is whether the arrested person is guilty Mathematically we can formulate the problem like this PO AN 2 1 PN 2 1 We need to condition as the experiment cannot even be performed when N 0 Now by the first Bayes formula POAlN21 POAN21 POAN2llNkPNk a l l H POAlNkPNk 3 48 k1 and 1 POAlNk so 13071 Ngt17 l A A 7 k 39 k 396 39 k1 The answer to the question then is 67A 00 A W WNPm m k1 There is no closed form expression for the sum but it can be easily computed nu merically The defense may claim that the probability of innocence 17the above probability is about 02330 when A l quite enough for a reasonable doubt This model was in fact tested in court in the famous People 2 Collins case a 1968 jury trial in Los Angeles In this instance it was claimed by prosecution on imsy grounds that p 112 000 000 and 11 would be the adult couples in the LA area say 11 5000000 The jury convicted the charged couple in a robbery on the basis of the prosecutors claim that due to low 1 the chances of there being another couple with specified characteristics in the LA area must be one in a billion77 The Supreme

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