# rank probability distribution in a round robin tournament for a fixed number of wins

Let suppose there are $n$ teams participating in a round robin tournament and a particular team has won $x$ matches out of total $n-1$ matches that it has played. Can you give me the probability distribution of that team with respect to rank considering ::

Part (1): Each team has $\frac12$ probability to win against any other team.

Part (2): Each team $i$ has an attribute $l_i$. So the probability of it winning against team $j$ will be $l_i/(l_i + l_j)$. So a stronger team will have higher $l_i$ and hence better probability of winning against teams with weaker $l$s.

Ties are broken randomly and each team plays every other team only once.

• point (2) sounds ..wrong: should it not be the probability to win $0,1,2,\cdots,n-1$ matches, instead ? – G Cab May 30 '18 at 18:08
• @ G Cab : Earlier my point was :: Team $i$ has an array $A_i$ of length n-1. Elements at index 1 to i-1 indicate probability to win against team 1 to team i-1. elements at index i to n-1 indicate probability to win against team i+1 to team n. $A_i[1]$ does not indicate probability to win first match as team i may not play its first match against team 1. But never mind I have simplified the question as new condition required fewer parameters and captures all the constraints I wanted to. – wondering mind May 30 '18 at 20:16
• Ok, now it is clear what you are looking for : a) "parts" 1& 2 are actually to be understood as "case" 1 and 2 b) case 2, can be put as that there is a matrix giving the probability of each team to win vs. others. Is that right ? – G Cab May 30 '18 at 20:31
• @ G Cab : yup, that's correct. – wondering mind May 30 '18 at 20:49