Probability histogram

A probability histogram draws one bar for each value a discrete random variable can take, and the height of the bar is $P(X = x)$. The heights add to exactly 1, because together they are the whole distribution. Equal-width bars sit centered on their values, so when the width is 1 each bar's area equals its probability as well and the total area is 1.

Roll a fair die three times and let $X$ count the sixes. The four bars have heights $\frac{125}{216} \approx 0.579$ at $x = 0$, $\frac{75}{216} \approx 0.347$ at $x = 1$, $\frac{15}{216} \approx 0.069$ at $x = 2$, and $\frac{1}{216} \approx 0.005$ at $x = 3$. They sum to 1, and the picture leans hard to the right.

That picture sets up the standard misreading: "the tallest bar is at 0, so the average number of sixes is 0." The tallest bar marks the most likely single value, which is a different question from the average. The mean here is $\mu_X = 0.5$ sixes, and it falls in the empty space between the bars at 0 and 1. Height reads likelihood; the balance point of the whole picture reads the mean, and on a skewed distribution the two are never in the same place.

Because the heights are probabilities, none can exceed 1 and all of them together make 1. A frequency histogram of data is not built that way: its heights are counts and they add to the sample size. A probability histogram also only exists for a discrete variable. A continuous one gets a density curve, whose vertical axis is density rather than probability and can rise above 1, since a uniform curve over an interval of width 0.5 has height 2.

One last thing the bars are not: data. They describe what the chance process does over many repetitions. Roll three dice twenty times and the histogram you actually get will not match these heights. It drifts toward them as the repetitions pile up.