Let $${U_1}$$ and $${U_2}$$ be two urns such that $${U_1}$$ contains $$3$$ white and $$2$$ red balls, and $${U_2}$$ contains only $$1$$ white ball. A fair coin is tossed. If head appears then $$1$$ ball is drawn at random from $${U_1}$$ and put into $${U_2}$$. However, if tail appears then $$2$$ balls are drawn at random from $${U_1}$$ and put into $${U_2}$$. Now $$1$$ ball is drawn at random from $${U_2}$$ being white is

Given that the drawn ball from $${U_2}$$ is white, the probability that head appeared on the coin is

Let $$P = \{ \theta :\sin \theta - \cos \theta = \sqrt 2 \cos \theta \} $$ and $$Q = \{ \theta :\sin \theta + \cos \theta = \sqrt 2 \sin \theta \} $$ be two sets. Then