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.

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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?