Intuitive explanation of why in binomial distribution the peak is reached at $x = np$ Suppose we toss $20$ coins with probability $p$. When we plot the graph, we see that the peak is reached at the p-th point in the graph, that is, around $x = 20p$ in this case. Can I get an intuitive (not with the binomial distribution formula) explanation why that is the case? Why is probability more at the point and decreases along a (smooth) curve on either sides. What exactly has the experiment number (which is $x$) got to do with this?
 A: The "peak" of the binomial distribution is also called the "expected value" or the binomial distribution.
Say you have a binomial experiment that will involve $n$ trials and each trial has a $p$ rate of success. Then what is the "expected number" of success's? It makes sense that the expected number of success's is proportional to both $n$ and $p$.
If we conduct more trials (increase $n$) then we should expect more success's. If we increase the success rate (increase $p$) then we should expect more success's. Thus the fact that the expected number of success's (the peak of the probability distribution) should equal $np$ makes sense intuitively.
The details on proving that this is the peak of the curve is $np$ comes from the the definition of the binomial distribution and expected value.
However, you incorrectly stated that the probability decreases on either side of the peak along a smooth curve. This is not true, since the binomial distribution is a discrete distribution, it's probability distribution will be "blocky" and not "smooth". That being said, for large $n$ the binomial distribution can be approximated by a normal curve, which is smooth.
