Often, news stories that are reported as startling "one-in-a-million" coincidences sure actually, upon closer examination, not rare events and can even be expected to occur. A few yesurs 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 (1989) call the "law of truly large numbers"? 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 of 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 part (b) still seems small, consider further that this event could have happened in any one of the 50 states, during any of the last 10 years, and still would have received the same news coverage.
In addition to Diaconis and Mosteller (1989), see Hanley (1992) for more on coinci¬dences.