How Sharp Is the Posterior?

A product manager models her belief about a landing page's conversion rate $\theta$ with a Beta distribution. Before running any test, her prior satisfies two conditions:

• its mean is 0.20;
• its two shape parameters sum to 15.

She then tests the page on 25 visitors, of whom 10 convert and the rest do not. The Beta distribution is the conjugate prior for Bernoulli trials, so her updated belief is again a Beta.

What is the standard deviation of her posterior distribution?

For $X \sim \text{Beta}(\alpha, \beta)$:

$E[X] = \frac{\alpha}{\alpha + \beta}, \qquad \text{Var}(X) = \frac{\alpha\beta}{(\alpha + \beta)^{2}\,(\alpha + \beta + 1)}$

Round to four decimal places.