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Birthday problem

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The birthday paradox states that if there are 23 people in a room then there is roughly a 50/50 chance that at least two of them have the same birthday. For around 60 or more people the probability is greater than 99%. This is not a paradox in the sense of it leading to a logical contradiction; it is a paradox in the sense that it is a mathematical truth that contradicts common intuition.

Calculating this probability (and related ones) is the birthday problem. The theory behind it was described in the American Mathematical Monthly in 1938 in Zoe Emily Schnabel's The estimation of the total fish population of a lake, under the name of capture-recapture statistics.

The key to understanding the birthday paradox is to realize that there are many possible pairs of people whose birthdays could match. Specifically, among 23 people, there are 23*22/2 = 253 pairs, each of which being a potential candidate for a match. To emphasize the point, consider a different scenario: if you enter a room with 22 people, the chance that somebody there has the same birthday as you is not 50/50, but much lower. This is because now there are only 22 possible pairs that could yield a match.

To compute the approximate probability that in a room of n people, at least two have the same birthday, we disregard leap years and twins, and assume that the 365 possible birthdays are equally likely. The trick is to first calculate the probability that the n birthdays are different. This probability is given by

p = 364 365 ⋅ 363 365 ⋅ 362 365 ⋯ 365 − n + 1 365 {\displaystyle p={\frac {364}{365}}\cdot {\frac {363}{365}}\cdot {\frac {362}{365}}\cdots {\frac {365-n+1}{365}}}

because the second person cannot have the same birthday as the first (364/365), the third cannot have the same birthday as the first two (363/365), etc. Using factorial notation, this expression can also be written as

p = 365 ! 365 n ( 365 − n ) ! {\displaystyle p={365! \over 365^{n}(365-n)!}}

Now, 1 - p is the probability that at least two persons have the same birthday. For n = 23 you will obtain a probability of about 0.507...

By contrast, the probability that someone in a room of n other people has the same birthday as you is given by

1 − ( 364 365 ) n {\displaystyle 1-\left({\frac {364}{365}}\right)^{n}}

which for n = 22 gives only about 0.059, and would need n to be at least 253 to give a value over 0.5.

The birthday paradox in its more generic sense applies to hash functions where the number of N-bit hashes you can generate before probably getting a collision is not 2, but rather only 2. This is exploited by birthday attacks on cryptographical systems.

How the birthday problem exemplifies bad effects of calculators

In his autobiography, Paul Halmos wrote:

"Hand-held calculators can be good things and they can have bad effects. The birthday problem can be used to exemplify a bad effect. A good way to attack the problem is to pose it in reverse: what's the largest number of people for which the probability is less than 1/2 that they all have different birthdays? .... the problem amounts to this: find the smallest n for which
∏ k = 0 n − 1 ( 1 − k 365 ) < 1 2 . {\displaystyle \prod _{k=0}^{n-1}\left(1-{\frac {k}{365}}\right)<{\frac {1}{2}}.}
The indicated product is dominated by
( 1 n ∑ k = 0 n − 1 ( 1 − k 365 ) ) n < ( 1 n ∫ 0 n ( 1 − x 365 ) d x ) n < ( 1 − n 730 ) n < e − n 2 / 730 . {\displaystyle \left({\frac {1}{n}}\sum _{k=0}^{n-1}\left(1-{\frac {k}{365}}\right)\right)^{n}<\left({\frac {1}{n}}\int _{0}^{n}\left(1-{\frac {x}{365}}\right)\,dx\right)^{n}<\left(1-{\frac {n}{730}}\right)^{n}<e^{-n^{2}/730}.}
The asserted domination comes from the celebrated relation between the geometric and arithmetic means; the next inequality comes from the definition of the definite integral, and the last one from 1 − x < e. .... The reasoning is based on important tools that all students of mathematics should have ready access to. The birthday problem used to be a splendid illustration of the advantages of pure thought over mechanical manipulation; the inequalities can be obtained in a minute or two, whereas the multiplications would take much longer, and be much more subject to error, whether the instrument is a pencil or an old-fashioned desk computer. .... What calculators do not yield is understanding, or mathematical facility, or a solid basis for more advanced, generalized theories. A pity."

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