A woman who was shopping in Los Angeles had her purse stolen by a young, blonde female who was wearing a ponytail. The blonde female got into a yellow car that was driven by a white male who had a...

A woman who was shopping in Los Angeles had her purse stolen by a young, blonde female who was wearing a ponytail. The blonde female got into a yellow car that was driven by a white male who had a mustache and a beard. The police located the blonde female named Janet Collins who wore her hair in a ponytail and had a friend who was a white male who had a mustache and beard and also drove a yellow car. The police arrested the two subjects.

Because there were no eyewitnesses and no real evidence, the prosecution used probability to make its case against the defendants. The probabilities below were presented by the prosecution for the known characteristics of the thieves.

Probability Distribution

1. Assuming that the characteristics listed are independent of each other, what is the probability that a randomly selected couple has all these characteristics? Tha is, what is P("yellow car" and "man with mustache" and "woman with ponytail" and "woman with blonde hair" and "white man with beard" and "man/woman in a car")?
   
   
   
   


2. Would you convict the defendants based on this probability?
   
   
   
   


3. Now let n represent the number of couples in the Los Angeles area who could have committed the crime. Let p represent the probability that a randomly selected couple has all six characteristics listed. Let the random variable X represent the number of couples who have all the characteristics listed in the table. Assuming that the random variable X follows the binomial probability function, we have P(x) = (_nC_x) * p^x * (1 - p)^n-x, where x = 0, 1, 2,..., n                                Assuming that there are n=1,000,000 couples in the Los Angeles area, what is the probability that more than one of them has the characteristics listed in the table (hint: 1-P(x=0)-P(x=1) by hand or in statcrunch binomial distribution calculator). Does this result cause you to change your mind regarding the defendants' guilt.