Birthday Probabilities
Why
Here’s a nice (surprising) example of computing an event probability. Consider the following question: In a group of $n$ people, what is the chance that there is a pair of people who were born on the same day of the same month.
Example
Suppose we model the sample space as the set
of all lists of length $n$ in the set $\set{1,
\dots , 365}$—ignoring leap years.
As usual, $\num{\Omega } = 365^n$.
Suppose further that we model all sequences as
equally likely.
In other words, we define a distribution $p:
\Omega \to [0,1]$ by
\[
p(\omega ) = \frac{1}{365^n}
\]
It is easier to think of the event that all
birthdays are distinct.
Then, using the properties of event
probabilities, we will be able to calculate the
probability of the complement of this event.
Define $D_n$ so that
\[
D_n = \Set{\omega \in \Omega }{\omega _1 \neq \omega _2 \neq
\cdots \neq \omega _n}
\] \[
P(D_n) = \frac{365\cdot 364\cdots(365-n+1)}{365^n}
\] \[
S_n = \Omega - S_n = \Set{\omega \in \Omega }{\exists i,j
\text{ with } \omega _i = \omega _j}
\] \[
P(S_n) = 1 - P(D_n) = 1 -
\frac{365\cdot 364\cdots(365-n+1)}{365^n}
\]
Even with small values of $n$, the above probability is quite large. For example, with $n$ at 23, $P(S_{23}) \approx .51$. For example, with $n = 40$, $P(S_{40}) \approx .89$. Many people refer to the result of this particular probabilistic model as the birthday paradox. The word paradox is used to indicate that the probability is higher than one might expect. The reasoning for this is that very few people know someone with their own same birthday, even though we know many ($>40$) people.
The resolution of the confusion is that if we
fix the day of the year, we can consider the
probability that in a group of $n$ people there
is an individual with the same birthday.
The probability of the appropriate event under
the same distribution as before can be shown to
be1
\[
1 - \frac{364^n}{365^n}
\]
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