People vs. Collins
Jury convicted on the basis of the small value. The California Supreme Court, however, argued as follows. Suppose we accept the value for P. Let n be the number of couples in the area. Let X = number of couples out of n who fit the description.
The Supreme Court argued that the relevant quantity here is not P, but the probability that there is some other couple that fits the description, i.e., P(X≥2/X≥1). We know X≥1, since Collins and her boyfriend fit the description.
Note that, for those binomial situations where n is very large and P is very small, the appropriate binomial probabilities(e.g., P(X=0), P(X=1), etc.) can be easily approximated by the Poisson function as follows:
P(X≥2/X≥1) depends upon the total number (n) in the Los Angeles area. Depending upon the quantity of n, the calculated probabilities are summarized in the table below:
“Reasonable doubt” was not been eliminated, according to this argument, and the verdict was overturned.