Binomial distribution for gender discrimination? I could be on the wrong path, but I'm trying to use binomial distribution to identify gender discrimination and am having issues getting consistent results based on trial size(?)... I think. Small numbers work as expected, large numbers do not.
An example that works: 16 people are hired. 2 are women. What is the chance this happened randomly? (x)=2, (n)=16, (p)=.5. Example below [from R] show what I expected.
pbinom(2, size=16, prob=0.5) 
#0.0021 -> .2% chance that there is no gender discrimination

An example that doesn't work: 1150 people are hired. 350 are women. What is the chance this happened randomly? (x)=350, (n)=1150, (p)=.5. Example below [from R] does not show what I expected.
pbinom(350, size=1150, prob=0.5) 
# 2.388788e-41 -> ?!

Thanks in advance. Any suggestions better approached please!

The results are consistent.
The differences in p-values that you get with trial size is as expected.
The dependency of the result on the parameter $n$ occurs because the distribution becomes more narrow.
The same relative deviation becomes less probable when $n$ increases.
See the below example (adapted from here) for the distribution of the observed number of women $k$ and the fraction of women $f$ as a function of the number of workers $n$.

Related question is How to estimate a probability of an event to occur based on its count?

Consistency
The fact that the same effect size (e.g ratio of women) becomes more significant is a good thing.
This relates to 'statistical consistency'*, and this is a desirable property. It means that we can make estimates better (as much as we want) by collecting larger samples.
So that is what you did with your sample of 1150, it is a more accurate estimate of the ratio of women than the sample of 16.

*An estimator is consistent if 'The probability for the estimator to differ by a certain quantity from the true value approaches zero, when we increase the sample size'.

Sidenote: Strictly speaking you only observed that the ratio of women is different from 0.5. But is this also discrimination?

Sidenote: This comparison of the 'deviation from the middle' is at the origin of p-values and the normal distribution. In 1710 similar comparisons were made by Arbuthnot comparing ratios in the birth of boys and girls. It is one of the first cases of expressing a p-value.
