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Re: Marilyn, again



John Mallinckrodt wrote:

I was, however, more interested initially in the response to
Marilyn's answer which I do disagree with. As Mark Shapiro points
out, 97.3% (I get 97.0% while Richard gets 97.5%) would indeed be
called "a sure thing" in many situations including, I dare say,
this one. (On the other hand, a 97% survival rate after taking
two aspirin for a headache might be thought of in distinctly
different terms.) In any event I see no reason to think that the
reader engaged in an "erroneous extrapolation" of the 23 person
result which might be expected to lead some to believe that
*absolute* (not virtual) certainty occurs with 46 (not 50) people.

Whether or not a 97+% probability of some event happening can be described as
a "sure thing", a "virtual certainty", or some other such expression of very
strong confidence is ultimately a subjective call. Clearly, if I had 1000
groups of 50 randomly selected people, it is not very certain that *all* 1000
of those groups would be found to have at least one matched birth date in
each group. In fact, it is highly likely that at least some of these groups
would have no matched birth dates. OTOH if I just had 1 sample of 50 people
(and never repeated the experiment) then I would be fairly confident of there
being at least one matched birth date in that group.

For the record if the assumption of a uniform distribution of birth dates
throughout the year is made, and if the year is assumed to have an integer
number of days in it, then the following formula obtains:

p = 1 - d!/((d-n)!*d^n)

Here p is the probability of there being *at least* 1 birthday match in the
group, n is the cardinality of the group, and d is the number of days in the
year. If we set d = 365 and n = 50 we get that p = 0.97037.... If we try to
crudely account for the possibility of there being a group member with a
Feb. 29th birthday by just interpolating the above formula for a noninteger
value of d = 365.2425 (length of the Gregorian year) we get p = 0.97030.
This is not much of a correction. Regarding the question of whether or not
the reader made an incorrect extrapolation of the n = 23 case to the n = 46
case to claim an absolute certainty, it seems to me that it is possible that
if the reader had made that proported extrapolation, he/she would have then
called it an 'absolute' certainty rather than a 'virtual' certainty. It is
possible that the reader merely meant that at the n = 50 group size the
probability exceeded the 95 % confidence level. It is interesting that the
95 % confidence level just happens to occur at n = 46 (actually,
at n = 46.273 if we are silly enough to allow for noninteger group sizes),
whereas at half of this value at n = 23, (or n = 22.775 to be silly) we have
the 50 % break-even even odds result.

David Bowman
dbowman@gtc.georgetown.ky.us