What is the correct interpretation of the probability in this scenario?

[mattmns]

I am having some trouble with the interpretation of the following question (specifically, part (a)).

Here is the question from the book.
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Often, news stories that are reported as startling "one-in-a-million'' coincidences are actually, upon closer examination, not rare events and can even be expected to occur. A few years ago an elementary school in New York state reported that its incoming kindergarten class contained five sets of twins. This, of course, was reported throughout the state, with a quote from the principal that this was a "statistical impossibility''. Was it? Or was it an instance of what Diaconis and Mosteller call the "law of truly large number''? Let's do some calculations.

(a) The probability of a twin birth is approximately 1/90, and we can assume that an elementary school will have approximately 60 children entering kindergarten (three classes of 20 each). Explain how our "statistically impossible'' event can be thought of as the probability 5 or more successes from a binomial(60, 1/90). Is this even rare enough to be newsworthy?

(b) Even if the probability in part (a) is rare enough to be newsworthy, consider that this could have happened in any school in the county, and in any county in the state, and it still would have been reported exactly the same. (The "Law of truly large numbers'' is starting to come into play.) New York state has 62 counties, and it is reasonable to assume that each county has five elementary schools. Does the event still qualify as a "statistical impossibility'', or is it becoming something that could be expected to occur?

(c) If the probability in (b) still seems small, consider further that this event could have happened in any of the 50 states, during any of the last 10 years, and still would have received the same news coverage.
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In part (a) I am not agreeing with the book that this probability can be thought of as 5 or more successes from a binomial(60, 1/90) distribution.

What I don't get is what is a success. We have 60 students, is a success the event that one of these students has a twin? If so, then there are actually two success because the twin, who I would assume is also in kindergarten at the same school, would also be a success. So I would think that we should be looking at the probability of 10 or more successes, not 5. If we had 5 successes then that would mean 5 of the 60 students have twins, which sounds right in some sense, but if the twins of these students are not in the 60 then where are they, and if they are in the 60 shouldn't there be 10, not 5?

Any thoughts on this? Am I missing something here, or did the book make a mistake?

Thanks!

 

[EnumaElish]

I think the book is correct "in the first order"; you are double counting.

Suppose there are n students in a classroom. We ask each student: "do you have a twin sibling?" If the the student answers "no," he or she leaves the classroom. If the student answers "yes," we ask the student to identify the sibling. Then both of them leave. If all students were to answer "no," the combinatorial coefficient would be 0Cn. If only one student answered "yes," then the combinatorial coefficient isn't 2Cn, it is 1C(n-1). If exactly k students answered yes, then the combinatorial coefficient isn't (2k)Cn, it is kC(n - k).