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The beta distribution lives on the interval from 0 to 1, which makes it the natural model for a probability or proportion. It is the conjugate prior for the binomial, so α and β act as counts of prior successes and failures.
Beta density
f(x) = x^(α−1)(1−x)^(β−1) / B(α, β)
Mean and mode
mean = α/(α+β), mode = (α−1)/(α+β−2) for α, β > 1
P(X ≤ 0.3) ≈ 0.5798, with a mean of 2/7 ≈ 0.2857 and a mode of 0.2.
Multiplying a beta prior by a binomial likelihood gives another beta, with the successes and failures simply added to α and β.