Counting Successes

Welcome back to The Binomial Distribution! In the first lesson, we identified the four conditions that must hold for a process to qualify as binomial: two outcomes per trial, a fixed trial count, a constant probability of success, and rough independence. Now, in this second lesson, we move past that checklist and ask a more interesting question: once we know a situation is binomial, what does its distribution actually look like, and how do we read it?

Here we shift from verifying conditions to interpreting results. We will see how every possible number of successes gets its own likelihood and how a bar chart makes those likelihoods visible at a glance. By the end, you will be able to look at a binomial bar chart and quickly identify which success counts are common and which are rare.

Imagine the binomial conditions are confirmed and we run our trials. Suppose we plant $n = 8$ seeds, each with a $p = 0.60$ chance of sprouting. After a week, we count how many sprouted. That count is a single number — anywhere from $0$ (none sprouted) to $8$ (all sprouted).

If we could repeat that entire batch of $8$ seeds many, many times, we would not get the same count every time. Some rounds might yield $4$ sprouts, others $6$, occasionally $2$ or even $8$. Each possible count would show up with a certain frequency. The binomial distribution is simply the pattern that tells us how likely each of those counts is. Let's see how that pattern takes shape.

When we run $n$ trials, the number of successes can be any whole number from $0$ to $n$. No fractions, no negatives, and nothing above $n$. For our $8$-seed example, the possible success counts are:

That gives us $n + 1 = 9$ possible outcomes in total. Every one of these counts has some probability attached to it, even if that probability is extremely small. The binomial distribution assigns a likelihood to each and every one of these counts, and together those likelihoods add up to $1$, since exactly one of the counts must occur.

The most natural way to display a binomial distribution is with a bar chart. The horizontal axis lists every possible success count from $0$ to $n$, and each bar's height shows the likelihood of that particular count.

Return to our $8$-seed scenario with $p = 0.60$. If we planted batch after batch and recorded the sprout counts, some counts would appear far more often than others. Plotting the results produces a chart where a few bars near the middle-to-right region are tall and the bars near the edges are short. The tallest bar marks the most likely count, while the shortest bars represent counts that almost never happen.

Because the possible outcomes are whole numbers (as we discussed in the very first course of this path, these are counted outcomes rather than measured ones), each outcome gets its own separate bar. There is no bar at $3.5$ or $6.7$ — only at $0, 1, 2, \ldots, 8$. This discrete, bar-by-bar structure is a defining visual feature of the binomial distribution.