4. In an alleged court case in Los Angeles in 1964, the prosecutor argued that t

Question

4. In an alleged court case in Los Angeles in 1964, the prosecutor argued that there was only a one in 12 million chance that a randomly selected couple such as the accused man and woman would have the characteristics observed at the scene of the crime (namely, a blond woman with a ponytail accompanied in a yellow car by a black man with a beard and a mustache.) Because the accused couple had these characteristics, they were convicted. (a) Assume there are 2 million male/female couples in the Los Angeles area. Let X be the number of these couples having the above characteristics. Assuming the prosecutor's 1 in 12 million figure is correct, what is the distribution of X? (b) Use the Poisson distribution to approximate the probability that at least one of the 2 million couples in Los Angeles has the above characteristics. Also approximate the probability that at least two couples have these characteristics (c) Calculate the conditional probability that there are two or more couples with these character- istics given that there is at least one (namely, the accused couple). Based on this calculation, the appeals court is said to have reversed the conviction

Explanation / Answer

(a) Here in the given problem, the distribution is binomial distribution with n = 2 million and p = 1/12

(b) Here if approximate the probability distribution we get expected number of such couples = 2 * 1/12 = 1/6

Pr(X >=1 ; = 1//6) = 1 - Pr(X =0) = 1 - e-1/6 = 1- 0.8465 = 0.1535

(c) Here we have to find the conditional probability that

Pr(there are two or more couples l there is at lease one such couple) = [1 - Pr(X =0) - Pr(X =1)] / [1 - Pr(X= 0)]

= [1 - e-1/6 - e-1/6 * (1/6)] / (1 - e-1/6) = 0.0124/ 0.1535 = 0.0808

so there are 8% chance of that thing to be happen.

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