programme evaluation: normal approximation to binomial

#1
Housing first is an approach to reducing homelessness. Supposedly, it has 80 percent success rate, which means that on average 80 percent of programme participants tend to retain their housing, and not fall back to homelessness.

It is also said that to reliably evaluate the effectiveness of this approach in any city, there must be a certain minimum number of programme participants. One city I know has selected 50 participants for their treatment group (and 100 for the control). The city claims that they have the necessary number of participants to provide reliable evidence after the end of programme whether the approach works or not.

I was wondering why the magic number was 50. Is it because of the normal approximation to the binomial, where the success-failure condition must each be equal or higher than 10?

np >=10
50 * 0.8 = 40 :tup:

n(1-p) >=10
50 * 0.2 = 10 :tup:

If so, then why exactly at least 10?