Question

In: Statistics and Probability

A woman who was shopping in Los Angeles had her purse stolen by a young, blonde...

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 man with a mustache and a beard. The police located a blonde female named Janet who wore her hair in a ponytail and had a male friend who had a mustache and beard and also drove a yellow car. Based on this evidence the police arrested the two suspects. Because there were no eyewitnesses and no real evidence, the prosecution used probability to make its case against the defendants. The probabilities listed below were presented by the prosecution for the known characteristics of the thieves. Characteristic Probability Yellow car 1/10 Man with mustache 1/4 Woman with ponytail 1/10 Woman with blonde hair 1/3 Man with beard 1/10 Interracial couple in a car 1/1000

(a) Assuming that the characteristics listed above are independent of each other, what is the probability that a randomly selected couple has all these characteristics? That is what is, calculate the probability: P( “yellow car” and “man w/ mustache, beard and … “interracial couple in car”)?

(b) Based on the above result would you convict the defendant? Explain thoroughly.

(c) 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 6 characteristics listed in the table. Assuming that the random variable X follows the binomial probability function, we have: ?(?) = ?(?,?) ∙ ? ? ∙ (1 − ?) ?−? , ? = 0, 1, 2, … ? Note: Use the calculator link http://stattrek.com/online-calculator/binomial.aspx Assuming there are n = 50,000 couples in the Los Angeles area, what is the probability that more than one of them has the characteristics listed in the table? ?(? > 1) =

(d) Does this result cause you to change your mind regarding the defendant’s guilt? Explain.

(e) The probability that more than one couple has these characteristics assuming there is at least one couple is given by the formula below and each is evaluated with the binomial formula from (c). ?( ? > 1 ∣ ? ≥ 1 ) = ?(? > 1) ?(? ≥1) = (f) Do you think the couple should be convicted “beyond all reasonable doubt” based on the answer from part (e)? Explain why or why not.

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orchestra answered 1 year ago
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