A common question with conversions and other rates, is how big a sample do you need to measure the conversion accurately? To get an estimate with standard error \(\sigma\) you need at most \( \frac{1}{4 \sigma^2} \) samples. In general if the true conversion rate is p it is \(\frac{p(1-p)}{\sigma^2}\).

So let's say we want to measure the conversion rate within about 5%. To be conservative we'd want the standard error to be a bit less than that, say 3% Then we would need at least \( \frac{1}{4 (0.03)^2} \approx 278 \) samples. Note that to double the precision we need to quadruple the sample.

If you want to run a null hypothesis test with 95% CI and 80% power then you need to multiply by the square of
2.8.
That is \( \frac{(2.8)^2}{4 \sigma^2} \), where \(\sigma\) is the detection size.
For an A/B test we require double this (since the variances add); so to see an uplift of 5% would require \( \frac{2 (2.8)^2}{4 (0.05)^2} = 1570 \) in *each* group (actually a little more due to the continuity correction.
If this seems too big maybe you don't actually want a significant test; look into something like test and roll.

For the details on why this is read from Bernoulli to the Binomial