Editing Birthday problem (section)

=== Number of days with a certain number of birthdays ===

==== Number of days with at least one birthday ====
The expected number of different birthdays, i.e. the number of days that are at least one person's birthday, is:
:<math>d - d \left (\frac {d-1} {d} \right )^n </math>
This follows from the expected number of days that are no one's birthday:
:<math>d \left (\frac {d-1} {d} \right )^n </math>
which follows from the probability that a particular day is no one's birthday, {{math|{{pars|s=150%|{{sfrac|''d'' − 1|''d''}}}}{{su|p=''n''|b=&nbsp;}}}}, easily summed because of the linearity of the expected value.

For instance, with {{math|1={{var|d}} = 365}}, you should expect about 21 different birthdays when there are 22 people, or 46 different birthdays when there are 50 people. When there are 1000 people, there will be around 341 different birthdays (24 unclaimed birthdays).

==== Number of days with at least two birthdays ====
The above can be generalized from the distribution of the number of people with their birthday on any particular day, which is a [[Binomial distribution]] with probability {{math|{{sfrac|1|''d''}}}}. Multiplying the relevant probability by {{mvar|d}} will then give the expected number of days. For example, the expected number of days which are shared; i.e. which are at least two (i.e. not zero and not one) people's birthday is:

<math display="block">d - d \left (\frac {d-1} {d} \right )^n - d \cdot \binom{n}{1} \left (\frac {1} {d} \right )^1\left (\frac {d-1} {d} \right )^{n-1} = d - d \left (\frac {d-1} {d} \right )^n - n \left (\frac {d-1} {d} \right )^{n-1} </math>