Evaluate $$\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx} $$

Urn $$A$$ contains $$6$$ red and $$4$$ black balls and urn $$B$$ contains $$4$$ red and $$6$$ black balls. One ball is drawn at random from urn $$A$$ and placed in urn $$B$$. The one ball is drawn at random from urn $$B$$ and placed in urn $$A$$. If one ball is now drawn at random from urn $$A$$, the probability that it is found to be red is ................

One hundred identical coins, each with probability, $$p,$$ of showing up heads are tossed once. If $$0 < p < 1$$ and the probability of heads showing on $$50$$ coins is equal to that of heads showing on $$51$$ coins, then the value of $$p$$ is

For two given events $$A$$ and $$B,$$ $$P\left( {A \cap B} \right)$$