I'm having difficulties to estimate the probability that two individuals are befriended with each other in a social network. Considering that out of a network of 153.111 people (knots), we recrutied 217 individuals. What are the odds that two individuals know each other in that network consisting of the 217 individuals? Can someone please give me a hint how to calculate this, as my probability skills are rather weak :-)
Thank you in advance.
There are $\binom{217}2=\frac{217\cdot 216}2=23436$ pairs of people. The probability that they all fail to know each other is $$(1/2)^{23436}$$ The probability that there is at least one pair that does know each other is then $$1-(1/2)^{23436}$$ This is very close to one. For instance $2^{10}=1024>10^3$ gives $2^{20000}>10^{6000}$ so the answer is $$0.999\dots $$ with at least 6000 nines before anything other than a nine!