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I did the experiment of asking 31 people
their birthdays, and I bet that would be a match,
was I taking a big risk?
Try out the experiment yourselves with this applet:
Supposing that the probability of
each day is
(which is not
quite true), we will use the complement trick,
is is easier to compute the probability of no
match rather than the probability of at least two matches,
in which all kinds of possibilities have to be enumerated.
I had a good chance of winning!
The general formula for k people in the room:
We use the approximation for x small that
so that this log is approximately
This approximation would have given