Thanks for the link to the blog post. It raised an important point worthy of inspection. I ran some numbers and "peeking" after the first 1000 trials does change the outcome. The chance that the outcome will reverse from declaring branch A the winner with 95% confidence to declaring branch B the winner with 95% confidence is rather small, less than 10%. However, if you lower your requirements to 80% confidence then the chance of the winner swapping increases to over 50%! For reference, I used the Wilson approximation for binomial distributions. I'm sure the Wald approximation fares worse.

Interesting, this is a weakness of significance testing, in particular, its parametric model. Using Bayesian inference you would be able to look early withou messing your results up.