Six heads from a random biased coin

EasyConditional Probability & Expectation Random Variables~4m

A bag holds biased coins whose heads-probability $p$ follows the density $f(p) = 2p$ on $(0,1)$ . You draw one coin at random and toss it $6$ times. What is the probability that all $6$ tosses are heads?

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Condition on the coin's bias $p$ : six heads has probability $p^6$ for that coin.
Average over the bias: $\int_0^1 p^6\,(2p)\,dp$ . Don't just use the mean bias.

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Six heads from a random biased coin

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