Expected triangles among five random teammates

Question

Five players share a ball during a game. Over the match, each pair of players ends up having passed to each other with probability $\tfrac12$ , independently across pairs. A trio is "well linked" if all 3 of its pairs passed to each other.

On average, how many well-linked trios are there among the 5 players?

Accepted Answer

Attach one indicator to each group of 3 players and sum. There are \binom{5}{3} = 10 trios. Fix one trio. It is well linked only if all 3 of its internal pairs passed, independently, each with probability 	frac12: P(	ext{a given trio is well linked}) = \left(	frac12ight)^3 = \frac18. By linearity over all 10 trios, E[	ext{well-linked trios}] = \binom{5}{3}\left(	frac12ight)^3 = 10 	imes \frac18 = \frac{10}{8} = \frac54 = 1.25. Takeaway. The formula \binom{n}{3}p^{3} counts expected triangles no matter the size: with n=5 and p=	frac12 it gives 	frac54. Overlapping trios are no obst…

Expected triangles among five random teammates

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