the shape of the binomial distribution
If you draw the bar chart of a binomial distribution — one bar for each possible count of successes from 0 to n — a clear personality emerges, and knowing it lets you sanity-check answers at a glance. The bars rise to a single peak and fall away, like a lopsided or symmetric mound. Where the peak sits and how lopsided the mound is are both controlled by just n and p.
Three facts shape the picture. First, the centre: the most likely count clusters near np, the mean. Second, the symmetry: when p = 0.5 the chart is perfectly symmetric (heads and tails are interchangeable), when p is small it leans right with a long tail toward larger counts (think rare successes), and when p is large it leans left. The skewness equals (1 - 2p) divided by the square root of np(1 - p), so it is zero at p = 0.5 and shrinks as n grows. Third, the spread: the standard deviation is the square root of np(1 - p), so the mound widens with more trials but the bars near the centre also get individually shorter because probability is spread over more outcomes.
The most beautiful consequence is that as n grows, the binomial bars trace out the familiar bell curve. This is the central limit theorem at work — a binomial is a sum of n independent 0/1 trials, and sums of many independent pieces go Normal. A common rule of thumb says the Normal approximation is decent once both np and n(1 - p) exceed about 10. But beware the other limit: if instead p shrinks while np stays modest, the bell never forms and you get the lopsided Poisson shape instead.
Binomial(10, 0.5) is symmetric, peaked at 5. Binomial(10, 0.1) is right-skewed, peaked at 1, with a long thin tail toward 5 or 6. Both are fully determined by n and p.
Symmetric at p = 0.5; skewed toward the rare side otherwise; bell-like as n grows.
The Normal approximation can suggest a positive probability for impossible counts (like negative or above n). A continuity correction helps, and for small p with modest np the Poisson shape is a better fit than the bell.