Six heads from a coin with a random bias

A coin is chosen at random from a large collection whose head-probabilities follow the density $f(t)=2t$ for $0<t<1$. You toss the chosen coin six times. What is the probability that all six tosses land heads?

$1/4$. Given bias $t$, six heads has probability $t^6$. Averaging over the density, $\int_0^1 2t\cdot t^6\,dt=\tfrac{2}{8}=\tfrac14$.