I am unable to convince myself intuitively as to why the variance of a binomial distribution increases with increase in n (number of trials). In general, I expect that as n increases, the distribution should converge near np = mean (p is the probability of success). We even define probability as a value which is obtained when n limits to infinity (i.e. n increases). So shouldn't I expect the when I increases n, the probability to get np=mean value increase as well?
Like in the picture above, as n increases, for constant p = 0.5, the variance increases.
Best Answer
Note that the values of $X \sim Binomial(n,p)$ correspond to number of "positive" trials, not probability. As $n$ grows, the values of $\hat{p} = X/n$ converge to the true $p$, hence the probability as a long-run frequency definition. Values of $X = n\hat{p}$ clearly don't converge.
Same distinction might help understanding the variance: variance of the proportion of positive trials decreases, $Var(X/n) = np(1-p) / n^2 = p(1-p)/n$, so the proportion estimates get more precise. Variance of the actual number of positive outcomes increases, however.