Please help to ...(this would need to refer to the article above)...
Judge Sullivean claimed that there was about 2 42% chance of finding a second such couple with the same characteristics as Janet and Michael Collins (refer to the article), show the calculation to support his statement.
(Assume that there are 12,000,000 couples in U.S.)
I think you might have problem to access the article, here is the article...
There are many applications of the use of probability and statistics in law. An interesting case involved a purse-snatching incident in the Los Angeles suburb of San Pedro. In 1964, Mrs. Juanita Brooks, 71, was knocked down on the street and had her purse containing $35 stolen. Mrs. Brooks was unable to positively identify the defendants in the case, Michael and Janet Collins, and it appeared at first that they would be acquitted of the charge against them.
The district attorney figured otherwise, however. It seems that a couple bearing certain characteristic had been seen running from the alley in which the mugging occurred, and these characteristics matched the Collins’ very closely. The D.A. brought in as an “expert” witness a mathematics professor from a local college who presented the following probabilities:
(1) Man with beard ……………. 1/10
(2) Blonde women ……………. 1/4
(3) Yellow car …………….. 1/10
(4) Women with ponytail …………….. 1/10
(5) Man with moustache …………….. 1/3
(6) Interracial couple in car …………….. 1/1000
Michael Collins was a black man with a moustache and a beard. Janet Collins was a blonde,
white women with a ponytail. They were married and owned a yellow car.
The mathematics professor argued that the correct way to determine the probability that given couple simultaneously possessed all six characters was to multiply the individual probability together. Hence, he reasoned, there was only one chance in 12,000,000 of finding such a couple. Since such a couple demonstrably did exist – the Collins were, after all, on trial – and since there was such a low probability if turning up such a couple by chance alone, it was likely, argued the D.A., that the Collins were, in fact, the guilty couple.
This was enough for the jury. Convinced by the “mathematics” they had seen, they duly convicted the Collins of the crime. It was, of course, appealed and this is where the story gets interesting. Judge Raymond Sullivan of the California Supreme Court wrote the court’s opinion overruling the lower court conviction of Janet and Michael Collins and citied three primary reasons, any of which would have been sufficient to reverse the lower court.
The first difficulty was that the professor had failed to give any demographic evidence for his probabilities (e.g., how do we know the probability of a yellow car is 1/10 ?). The figures given were, in court view, little more than wild conjecture and certainly not grounds for convicting a couple on circumstantial evidence.
The second objection was a mathematical one, regarding the use of multiplication rule in probability theory. According to Judge Sullivan the use of multiplication rule in the calculation was inappropriate.
The third point was the most interesting, using the theory of conditional probability, Judge Sullivan mathematically demonstrated that is was quite likely that another couple with the same six characteristics existed, given that one such couple existed. Using 1/12,000,000 figure given by the D.A., Judge Sullivan showed that there was about a 42% chance of finding a second such couple with the same characteristics as Janet and Michael Collins. That is even though there was only 1/12,000,000 chance of turning up one couple with six characteristics, there is a 2/5 probability of turning up a second such couple once you’ve found the first. The judge’s reasoning was correct and appears on the court transcript of the trial.
(Source:Basic Statistics, by Kiemele & Schmidt © 1993 Air Academy Press)
anyone can help me...
Thank you very much...
Let n = 12,000,000. The probability of finding a couple with the six characteristics was assumed to be 1/n. Assuming as in your first post that the number of couples is n, the number of couples with the six characteristics has a binomial distribution with success probability p = 1/n and sample size n.
Originally Posted by eureka
Let k be the number of couples with the six characteristics. Compute the binomial probabilities
P(k = 0) = B(0;p,n) = (1-p)^n.
P(k = 1) = B(1;p,n) = np(1-p)^(n-1).
P(k >= 1) = 1 - P(k = 0).
P(k >= 2) = 1 - P(k >= 1).
Let A be the event that at least one couple exists with the six characteristics. Let B be the event that two or more such couples exist. P(A) = P(k >= 1). P(B) = P(k >= 2). P(A and B) = P(B) since if two couples exist, then at least one exists.
The conditional probability of finding 2 couples with the six characteristics given that at least one such couple exists is, from the definition of conditional probability,
P(A and B)/P(B) = P(B)/P(A) = P(k >=2)/P(k >= 1) = .42 = 42%
after calculating those probabilities as described above.
Thank you so much for your help.. i have another question;
For the second objection, Judge Sullivan said that the use of multiplication rule in calculation was inappropriate (refer to the article), why did he say so? Give a reason to support his statement.
Do you have any idea?
Using the multiplication rule to compute probabilities requires that the events be independent.
Originally Posted by eureka