Ten white draws from coin-coloured balls

A bag holds ten balls. Each ball was painted black or white according to a separate fair coin toss. You then draw a ball, note its colour and replace it, ten times in total, and every draw is white. What is the probability that all ten balls in the bag are white?

About $0.070$. The number of white balls $K$ is $\text{Bin}(10,\tfrac12)$ a priori and ten white draws have likelihood $(K/10)^{10}$. Bayes gives $\dfrac{1}{\sum_{k=0}^{10}\binom{10}{k}(k/10)^{10}}\approx0.0702$.