I am working with following Principle (My Approach At Bottom) :
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 periodically,
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 105 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 105,
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 n2/2. We shall use this approximation in what
follows. Thus, the number of pairs of people is 109
2 = 5 × 1017. The number
of pairs of days is 1000
2 = 5 × 105. 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
5 × 1017 × 5 × 105 × 10−18 = 250, 000
My Approach
One person visiting a particular hotel on given day = 1 / 100 * 100000
Two person doing above is square of above no(DC).
Total no of comb of person and pair of hotel and pair of days (TC) = Billon choose 2 * No of hotel choose 2 * No of days choose 2
So No of people under scrutiny = DC * TC
My question is why in the solution author choose just to divide by no of hotel once not twice as there are two people.
It is big statement problem. Thanks for going through.
Vishal
Best Answer
Given any particular hotel, the probability they both visit that hotel is indeed obtained by dividing by $(10^5)^2$. But then we must add up over all the $10^5$ hotels. Effectively that leaves a single $10^5$ in the denominator.
Here is another way to look at it that is perhaps more intuitive. Whatever hotel Alicia visits, the probability that Beti chooses the same one is $\dfrac{1}{10^5}$.