I am reading a chapter of book Mining of Massive Data Sets
book is available here
http://www.mmds.org
Chapter 1 http://infolab.stanford.edu/~ullman/mmds/ch1.pdf
Now in Section 1.2.3 An example of Bonferroni's principle has been given.
I have not been able to understand following text from book
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
How do they come to conclusion that there are 100,000 hotels?
How are 100,000 hotels enough to hold 1% of a billion people.
Then in the same example they have mentioned following
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.
This probability calculation is also not clear to me. Any help will great.
Thanks
Best Answer
We are studying $10^9$ people.
On any day, we should expect $1\%$ of them to visit a hotel, which would be $10^9 \cdot 10^{-2}=10^7$ of them deciding to visit a hotel.
Each hotel can hold $100=10^2$ people, to handle then $10^7$ people, we need $\frac{10^7}{10^2}=10^5$ hotels in the market.
Now for your second question, the probability that two particular people would decide to visit some hotel on the same day independently is $(10^{-2})^2=10^{-4}$. Remember that we have $10^5$ hotels, hence the chance of visiting the same hotel on the same day would be $(10^{-5})(10^{-4})=10^{-9}$. The probability that they will visit the same hotel on $2$ different day would then be $(10^{-9})^{2}=10^{-18}$.