Success Chance and Shape

Welcome back to The Binomial Distribution! So far in this course, you have learned what makes a process binomial and how to read the resulting bar chart — finding the peak, scanning the likely range, and spotting the rare counts in the tails. Now, in this third lesson, we ask a natural follow-up question: what happens to that bar chart when we change the probability of success?

Here we will explore how the success chance $p$ reshapes the entire distribution. Adjusting $p$ shifts where the bars pile up, tilting the distribution toward few successes, toward many, or balancing it right in the middle. By the end, you will be able to predict the shape of a binomial distribution just by knowing $p$.

Recall from the previous lesson that the peak of a binomial bar chart tends to sit near $n \times p$. We saw this with $8$ seeds at $p = 0.60$, where the peak landed around $5$, and with $10$ light bulbs at $p = 0.90$, where it landed at $9$. In both cases, we held $p$ fixed and read the chart. But what if we keep $n$ fixed and vary $p$ instead?

Think of $p$ as a control knob that slides between $0$ and $1$. Turning the knob changes where the bars concentrate. A low setting crowds the bars toward the left (few successes), a middle setting centers them, and a high setting pushes them to the right (many successes). Let's walk through each scenario, all using $n = 10$, so the only thing changing is $p$.

The comparison below shows this shift side by side.

Suppose we flip $n = 10$ coins, but each coin has only a $p = 0.10$ chance of landing heads. Out of $10$ flips, we expect about $n \times p = 1$ head on average. That single number already tells us a lot about the shape: the tallest bars will cluster near $0$ and $1$, and counts like $7$, $8$, or $10$ will be almost impossible.

On the bar chart, the distribution leans heavily to the left. The bar at $0$ is tall, the bar at $1$ is the tallest or nearly so, and every bar after that drops off quickly. The right side of the chart is practically empty. You may notice that this creates a lopsided shape with a long, thin tail stretching to the right — a pattern called right-skewed (named after the tail direction, not where the bulk sits). When $p$ is small, successes are hard to come by, so the bars bunch up near few successes.