The probability of India winning a test match against West Indies is $$1/2$$. Assuming independence from match to match the probability that in a $$5$$ match series India's second win occurs at third test is

Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral, equals

$$\,3{\left( {\sin x - \cos x} \right)^4} + 6{\left( {\sin x + \cos x} \right)^2} + 4\left( {{{\sin }^6}x + {{\cos }^6}x} \right) = $$

The value of $$\int\limits_\pi ^{2\pi } {\left[ {2\,\sin x} \right]\,dx} $$ where [ . ] represents the greatest integer function is