# The security of Zero knowledge proof performance

I recently learned to study the security of zero knowledge proof. As you can see on the wiki (link), it seems to be the most popular example of the Ali Baba cave. I have a question about the security of ZKP (Zero knowledge proof) in the Ali Baba cave. The reasons are as follows: During the verification process, the prover will deliver the value of $$e = [0,1, \ldots]$$. And $$e$$ will have a solution to either left or right depending on $$0$$ or $$1$$. And according to the length of list $$e \space (2 ^ n)$$, this process is said to be more secure. However, there are only two paths, and depending on the value of $$e$$, the secret key value is not changed in directions $$A$$ and $$B$$. So, should not the safety of this algorithm be just $$2$$ ?

For example, to solve 4 challenges the adversary has to choose the side door $$4$$ times, so it has the probability $$\frac{1}{2^4}$$ to win the game.