Birthday Type Problems
The original birthday problem
What is the probability that in a group of 30 people there are (at
least) two that share the same birthday?
A naive, but incorrect
argument would lead one to expect something in the
neighborhood of 30/365 or about 8%.
Suprisingly, the probability is much larger than this, namely 70 %.
For a group of 60 people, this probability rises to 99.4 %; i.e., it is
almost certain that two out of those 60 people have the same birthday
even though these 60 people collectively represent at most one out of 6
possible birthdays.
Since these probabilities are unexpectedly high
and rather counterintuitive, the problem is also known
as the birthday paradox.
For a nice java simulation of the birthday problem click
here.
Variations on the birthday theme
Marilyn vos Savant, who holds the distinction of having the highest IQ
in the world, discussed the birthday problem in her column in Parade
Magazine.
The following problems are reader replies to this column and
appeared in Parade Magazine, 11 Jan. 1998, p. 8.
While the problems involve birthdays and, on the surface,
appear to be similar to the original birthday problem,
not all are mathematically similar to the birthday
problem, and some (e.g., Problem 2) require a completely
different argument. It is essential that you see beyond
the superficial similarity of these problems and spot the
mathematical differences between the various problems.
- Problem 1:
I was born on Jan. 30. When my wife next went
to the mall, I tagged along to take you up on
the random survey. I spoke with 100 people, and
28 gave me the cold shoulder. Of the remaining 72,
no one shared Jan. 30 as a birthday. What happened?
Discussion Questions:
- Why should the reader not be surprised about this outcome
(namely that none of the 72 people shared the reader's birthday)?
- What is the probability that this outcome occurs?
- Problem 2:
I started counting from the first person who came
came into the office. I counted until I found a
matching birthday in the group. Then I started a new
survey with the next person. In eight surveys, the
smallest number of people it took before I found
a matching birthday was 12. The largest number was
only 54.
Discussion Questions:
- Is the minimum of 12 surprising? Is the maximum of 54 surprising?
- What is the probability that, in a single survey
one has to ask less than 12 (or more than 54) in order to obtain a
matching pair of birthdays?
- What is the probability that in at least one of eight such surveys
the number is less than 12?
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Last modified Tue 11 Feb 2003 03:10:53 PM CST