Databases Reference
In-Depth Information
1.2.3 An Example of Bonferroni's Principle
Suppose there are believed to be some “evil-doers” out there, and we want
to detect them. Suppose further that we have reason to believe that periodi-
cally, evil-doers gather at a hotel to plot their evil. Let us make the following
assumptions about the size of the problem:
1. There are one billion people who might be evil-doers.
2. Everyone goes to a hotel one day in 100.
3. A hotel holds 100 people. Hence, there are 100,000 hotels - enough to
hold the 1% of a billion people who visit a hotel on any given day.
4. We shall examine hotel records for 1000 days.
To find evil-doers in this data, we shall look for people who, on two different
days, were both at the same hotel. Suppose, however, that there really are no
evil-doers. That is, everyone behaves at random, deciding with probability 0.01
to visit a hotel on any given day, and if so, choosing one of the 10
5
hotels at
random. Would we find any pairs of people who appear to be evil-doers?
We can do a simple approximate calculation as follows. The probability of
any two people both deciding to visit a hotel on any given day is .0001. The
chance that they will visit the same hotel is this probability divided by 10
5
,
the number of hotels. Thus, the chance that they will visit the same hotel on
one given day is 10
−9
. The chance that they will visit the same hotel on two
different given days is the square of this number, 10
−18
. Note that the hotels
can be different on the two days.
Now, we must consider how many events will indicate evil-doing. An “event”
in this sense is a pair of people and a pair of days, such that the two people
were at the same hotel on each of the two days. To simplify the arithmetic, note
that for large n,
n
2
is about n
2
/2. We shall use this approximation in what
follows. Thus, the number of pairs of people is
10
9
2
= 5×10
17
. The number
1000
2
= 5×10
5
. The expected number of events that look
like evil-doing is the product of the number of pairs of people, the number of
pairs of days, and the probability that any one pair of people and pair of days
is an instance of the behavior we are looking for. That number is
of pairs of days is
5×10
17
×5×10
5
×10
−18
= 250, 000
That is, there will be a quarter of a million pairs of people who look like evil-
doers, even though they are not.
Now, suppose there really are 10 pairs of evil-doers out there. The police
will need to investigate a quarter of a million other pairs in order to find the real
evil-doers. In addition to the intrusion on the lives of half a million innocent
people, the work involved is su
ciently great that this approach to finding
evil-doers is probably not feasible.
