I may not be fully understanding your correction here, but it sounds like you are unnecessarily complicating the calculation into one in which it becomes a ‘no-replacement’ problem (hence your need for combinations). But in reality these last digits are repeatable (and indeed must repeat since we have 30 instances and only 10 digits to choose from). It would be natural to assume that each last digit is independent from the other last digits – indeed this is a required assumption for these kinds of calculations. So (on third thought?) – the simpler original calculation seems correct to me: (1/2) ^ 30 comes to about one in a billion.
My correction answers the question: “how remarkable is it that the last digit of the ages of 30 patriarchs are limited to a set of 5 possibilities?!”
It is true that “how remarkable is it that the last digit of the ages of 30 patriarchs are limited to the set {0,2,5,7,9}?” is less complex. It is also the wrong question to ask, in my opinion. One response to this second, less complex question would be: “So what? It might be improbable that the Powerball lottery today picked the particular set of 6 numbers that got read on the news, but it’s not improbable that 6 numbers got picked. In the same way, it might be improbable that the ages all end with those particular 5 digits, but maybe it’s not improbable that the last digits would all be selected from a set of 5.”
My calculation answers this very legitimate objection. So what if it’s more complex? It’s not unnecessarily complex.
EDIT: Here’s a Richard Feynmann story that illustrates my point. When Feynmann would give a lecture, he would often say something like this, “I just saw a car with the license plate B-R-U-2-9-8 on my way to the lecture hall. What are the odds?!”
Of course, the odds of seeing that particular combination of letters and numbers are indeed only about 1 in 10 million–infinitesimally low. But Feynmann’s parody points out why that probability is irrelevant: the right question to ask is: “What are the odds of seeing a license plate with three letters followed by three numbers?” And those odds are very, very high indeed.
In the same way, asking the probability of seeing the last digits of 30 ages drawn from a particular set of 5 is the wrong question to ask. It is no more relevant than asking the odds of seeing a particular combination of letters and numbers on a license plate. A better, more relevant question is, “What are the odds of seeing the last digits of 30 ages restricted a set of 5?”
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Okay – now I see (and agree with) where you’re coming from. Thanks for the extra clarification! Always good to see the math done right.
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