I'm new to calculating probabilities, so I need my math checked on this.
I have taken a sample of 7 of a population of 2550 on a binomial test. First, I wanted to see if there are only 3 yeses in a 2550 population, what are the odds that I selected all three in a sample of 7. I used C(2550,7)/C(840,1).. or 1.38x10^20/840. I hope that is the correct math?
$n=2550$
$r=7$
$C(2550,7)/C(840,1) = 1.38(10)^{20}$
Given such a low probability, and not knowing the true number of yeses, I'm hoping to draw some high-threshold conclusions about my small sample of 7. If my sample shows 42.9% yeses, what is my 99% lower level confidence interval? z = 2.33. Mean of the bimodal is .4286, stdev is .5345. That gives me +/- 47.1%, or outside the bounds. Am I looking at this right?
$n_,=7$
$\mu=.4286$
$s_1=.5345$
$z=2.33$