The shape is set by two numbers, and a handy way to read them is this: $\alpha$ counts the successes you have seen (plus one) and $\beta$ counts the misses (plus one). A player who sinks 3 and misses 1 gives Beta$(4,2)$. Your single best guess for her skill is the mean, $\alpha/(\alpha+\beta)=4/6\approx0.67$. Watch more shots and both dials climb together: the curve narrows and your confidence grows.

The density is $f(x;\alpha,\beta)=\dfrac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$ on $[0,1]$, where $B(\alpha,\beta)=\dfrac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}$ is just a constant that rescales the area to exactly $1$. The mean is $\alpha/(\alpha+\beta)$, and the spread shrinks roughly like $1/(\alpha+\beta)$ — more total evidence makes a tighter curve. Beta is also the conjugate prior of the binomial: start with Beta$(\alpha,\beta)$, observe $k$ successes in $n$ trials, and your updated belief is exactly Beta$(\alpha+k,\;\beta+n-k)$. Bayesian updating is nothing more than adding to the dials. In the sim, Parameter 1 is $\alpha$, Parameter 2 is $\beta$, and Query x reports the area to the left of $x$ — the probability that the true rate is below that value.

Set $\alpha=\beta=1$ and watch the curve flatten into a straight line — the "I know nothing yet" uniform prior. Now drag both up to about $10$ for a tight bell over $0.5$, the look of strong evidence. Finally set $\alpha=\beta=0.5$ to get a U-shape that piles probability near $0$ and $1$ — the belief that a coin is biased one way or the other, you are just not sure which.

This distribution has important theoretical and applied properties. See the simulation tabs to explore how shape changes with parameters.

Bayesian statistics (conjugate prior for binomial p), proportion modeling, A/B test click rates, task completion percentages. The Beta distribution is "the distribution for probabilities" — it lives on [0,1] and can take many shapes: uniform (α=β=1), bell (α=β>1), U-shaped (α=β<1), skewed, or J-shaped.