# Probability of same common variable from select size pool

I've been playing a game recently where two teams of 5 random characters from a pool of 75 are pitted against each other. I've noticed that surprisingly often both teams seem to have the same characters. This got me thinking about how likely this is and since probability has never been my forte I began searching for an answer but was unsuccessful. So I was curious if anyone would have any idea of how to go about calculating this?

General format: if 2 parties pick K random variables from N choices. What is the chances they will have at least one in common?

Thanks!

• If I understand correctly, by "both teams have the same characters", you mean "the two teams have at least one character in common"? – Sambo Jul 11 at 14:49

After the first person choose $$K$$ characters, the second can choose a disjoint set of $$K$$ characters in $${N-K\choose K}$$ ways. The probability that the two sets intersect is $$1-{{N-K\choose K}\over{N\choose K}}$$ When $$N=75,\,K=5$$ this is $$1-{70!70!\over75!65!}=1-{70\cdot69\cdot68\cdot67\cdot66\over75\cdot74\cdot73\cdot72\cdot71}\approx0.2987577.$$