First of all, your calculations on the number of pairs of people are wrong. There are 100 000 000 people, that's ${10}^{8}$, which gives us $5 \times {10}^{15}$ pairs of people (using the formula you wrote).

The probability of two people buying the same set of objects is ${1 \over 10^{23}}^2$ - because the same set has to be picked twice.

At the end we would get $100 \times (5 \times 10^{15}) \times ({1 \over 10^{23}})^2 ≈ 5 \times (10^{17}) \times {1 \over 10^{46}} ≈ {5 \over 10^{29}} ≈ {1 \over 2\times10^{28}} ≈ {1 \over 10^{28}}$ (I dropped the 2.6 just to make the equation simpler)

The main question here is:


  "If we search for pairs of people who have bought the same set of items, would we expect that any such people found were truly criminals?"


The answer is yes, because:


  "The Bonferroni principle says that we may only detect terrorists by looking for events that are so rare that they are unlikely to occur in random data" 


and our event is very unlikly to happen.

The Probability of Catching Criminals
How can I enable the PDF bar in Google Chrome?

I can view my PDF files, but recently I can't use all the buttons such as Save, Print... my bar disappeared.
