Other birthday problems
The birthday problem has been generalized in many ways. One approach is to
consider three people sharing a birthday. In this case 88 people would be
required before there is a better than even chance that three people will share the
same birthday. There are correspondingly larger groups if four people, five
people,... are required to share a birthday. In a gathering of 1000 people, for
example, there is a better than even chance that nine of them share a birthday.
Other forays into the birthday problem have inquired into near birthdays. In
this problem a match is considered to have occurred if one birthday is within a
certain number of days of another birthday. It turns out that a mere 14 people in
a room will give a greater than even chance of two people having a birthday in
common or having a birthday within a day of each other.
A variant of the birthday problem which requires more sophisticated
mathematical tools is the birthday problem for boys and girls: if a class consists
of an equal number of boys and girls, what would be the minimum group that
would give a better than even chance that a boy and a girl shared a birthday?
The result is that a class of 32 (16 girls and 16 boys) would yield the
minimum group. This can be compared with 23 in the classic birthday problem.
By changing the question slightly we can get other novelties (but they are not
easy to answer). Suppose we have a long queue forming outside a Bob Dylan
concert and people join it randomly. As we are interested in birthdays we may
discount the possibility of twins or triplets arriving together. As the fans enter
they are asked their birthdays. The mathematical question is this: how many
people would you expect to be admitted before two consecutive people have the