Comparing 2 means (binomial distribution type) between 2 groups

Hi, I feel stupid for asking this, but I have a basic "why" question about obtaining the mean values to later run a 2-sample t-test to see if 2 groups are significantly different. I've read online that for binomial distributions (ie yes/no answers, successes/failures, etc) the mean = np, where n is the sample size, and p is the probability of successes.

My question is how the mean can be so dependent on n, where if n was 1000 or 10, it can alter the mean by a factor of 100! This is in contrast to say weight, where whether you average the weights of 10 ppl vs 1000 ppl, the means should be close to one another. More on my confusion below:

Ex 1
n group 1 = 1000
n group 2 = 10
% success (or p) group 1 = .0035
% success (or p) group 2 = .35
mean group 1 = 3.5
mean group 2 = 3.5

On its surface, even before considering sd, p-value, etc., how can the means possibly be the same, when the % success is so different?

Ex 2
n group 1 = 1000
n group 2 = 10
% success (or p) group 1 = .35
% success (or p) group 2 = .35
mean group 1 = 350
mean group 1 = 3.5

Since the relative % success is the same, shouldn't the means be the same or similar?



TS Contributor
"mean"means mean number of occurrences, so having a low probability event, but trying many times should give you roughly the same number of succeses as a higher probability event with less trials.

Thank you rogojel. So I guess my question now is how you can use a T test to compare whether the 2 groups in example 2 that I showed above, are from the same population, when their means are so wildly different? But they should be from the same population theoretically (since they have similar proportion of successes), just that one was sampled 10 times while the other 1000 times.

continuing with example 2 from first post:
n group 1: 1000
n group 2: 10
% success group 1: .35
% success group 2: .35
mean group 1: 350
mean group 2: 3.5
sd group 1: rad (npq) = rad (350*.35*.65) = 8.92
sd group 2: rad (npq) = rad (3.5*.35*.65) = 0.892

So how could group 1, with a mean of 350 and sd of 8.92, be considered from the same population as group2, with a mean of 3.5 and sd 0.892? Or is the conclusion that these are significantly diff populations (even though that's a function of only sample size)?

I mean running any 2-sample t-test based off means and SDs is going to show these are VERY diff populations even though I don't think that's the case. Or is this the wrong test?