Very simple poker holdem question N holdem hands (just 2 cards from a standard 52 cards deck)  are dealt to N players
How to compute the probability that:
-exactly k players have a pair ( Two cards of the same value e.g.: 7, 7).
-at least k players have a pair

The problem seems simpler that I thought.The probability that any player has a pair is constant (3/51=0.0588). Hence,if I am right:


*

*the probability that k players have a pair is given by the binomial distribution B with parameters N and 0.588: P(K=k)=B(N,0,0588)

*the probability that at least k players have a pair is given by the cumulative
1-I(1-0.0588)(N-k,1+k) where I is the regularized incomplete beta function

