# Question on the chi-square test and Poisson distribution for significance testing

A quick question to check some reasoning: consider the chi-square test for a 2x2 table. For sake of argument, there's 500 patients in each group / category (male and female) and 10% of the total have some condition - if that consists of 60 men and 40 women with the illness, a chi-squared test gives a statistic of 4.444, significant at $$\alpha < 0.05$$ indicating a difference between men and women. Let's consider what happens if this disease were Poisson distributed, with expected value $$\lambda = 50$$ cases per group. Then we can simulate the probability of finding $$k$$ cases in a subset of people, and it looks like this:

The reason this is throwing me a little is that the areas under the tails are more than I'd expect at $$\alpha = 0.05$$; for example, a chi-square test suggests you'd find significance if men had $$k\leq 40$$ and $$k\geq 59$$ cases, but the sum of the area under the curve between $$0-40$$ and $$59-100$$ seems quite substantial (quick calculation suggests one has about ~18% chance of lying in this region). I'm wondering if the reason for this is that I can't just consider one group in isolation, and have to multiply probability - for example, if I see 60 cases in men, I must see 40 cases in women? Or is there some reason Poisson assumptions (or similar) don't hold here? Any insight welcome, just to make sure I properly understand what's going on....