A fair coin is tossed repeatedly. If the tail appears on first four tosses, then the probability of the head appearing on the fifth toss equals

For the three events $$A, B,$$ and $$C,P$$ (exactly one of the events $$A$$ or $$B$$ occurs) $$=P$$ (exactly one of the two events $$B$$ or $$C$$ occurs)$$=P$$ (exactly one of the events $$C$$ or $$A$$ occurs)$$=p$$ and $$P$$ (all the three events occur simultaneously) $$ = {p^2},$$ where $$0 < p < 1/2.$$ Then the probability of at least one of the three events $$A,B$$ and $$C$$ occurring is

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