BB84 - Bits needed to detect Eve's presence

From Quantum Computing: A Gentle Introduction:

In the BB84 protocol, how many bits do Alice and Bob need to compare to have a 90% chance of detecting Eve's presence?

I am having some trouble calculating this value. I understand that 50% of the time, Eve measures in the wrong basis and sends Bob a different bit than Alice sent, and after Alice and Bob compare their measuring bases, on average, Bob is left with 50% of his measured qubits. Therefore, given that Eve eavesdropped, there is a 25% probability that a given remaining bit is wrong. From this information, how would I calculate the number of bits needed to detect Eve with 90% confidence?

• I thought about this some more and came up with the following (if someone can confirm that would be great): for each bit that Alice and Bob compare, there is a 75% chance that they do not see a wrong bit. They want to get the probability of seeing this matching bits given Eve eavesdropped to less than 10%. Therefore, we have $0.75^x < 0.10$, where $x$ is the number of bits to compare, leaving a final solution of ~ 8 bits needed to detect Eve's presence with 90% confidence. Aug 3, 2016 at 11:51