Alice, Beatrice and a tournament In a tournament of $2^n$ players, Alice and Beatrice ask what's the probability that they'll not compete if they've the same level of play? Let : * $A_i$ : Alice plays the $i$-th tournament ; * $B_i$ : Beatrice plays the $i$-th tournament ; * $E_i$ : Alice and Beatrice don't compete at the $i$-th tournament. * * * For the moment, I was only able to calculate $$P(A_i) = \left(\frac{1}{2}\right)^{n-1} \quad \forall^{\;i}_{\; 1 \dots n}$$ Can you give me a hint?

Hints:

Assuming every result is equally probable,

* How many contests are there? (All but one of the competitors need to be knocked out)

* How many potential pairings are there?

* What proportion of potential pairings actually meet in a contest? Can you simplify this?