Appendix A
The Birthday Problem
The setting is that we have
q
balls. View them as numbered, 1
, . . . , q
. We also have
N
bins, where
N
≥
q
. We throw the balls at random into the bins, one by one, beginning with ball 1. At random
means that each ball is equally likely to land in any of the
N
bins, and the probabilities for all the
balls are independent. A collision is said to occur if some bin ends up containing at least two balls.
We are interested in
C
(
N, q
), the probability of a collision.
The birthday paradox is the case where
N
= 365.
We are asking what is the chance that, in a
group of
q
people, there are two people with the same birthday, assuming birthdays are randomly
and independently distributed over the days of the year.
It turns out that when
q
hits
√
365 the
chance of a birthday collision is already quite high, around 1
/
2.
This fact can seem surprising when Frst heard. The reason it is true is that the collision probability
C
(
N, q
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 Spring '05
 HUANG
 Collision, Probability theory, ball, Birthday attack, Birthday problem

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