A lot contains $$50$$ defective and $$50$$ non defective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events $$A, B, C$$ are defined as
$$A=$$ (the first bulbs is defective)
$$B=$$ (the second bulbs is non-defective)
$$C=$$ (the two bulbs are both defective or both non defective)
Determine whether
(i) $$\,\,\,\,\,$$ $$A, B, C$$ are pairwise independent
(ii)$$\,\,\,\,\,$$ $$A, B, C$$ are independent

In a test an examine either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he make a guess is $$1/3$$ and the probability that he copies the answer is $$1/6$$. The probability that his answer is correct given that he copied it, is $$1/8$$. Find the probability that he knew the answer to the questions given that he correctly answered it.

A is a set containing $$n$$ elements. $$A$$ subset $$P$$ of $$A$$ is chosen at random. The set $$A$$ is reconstructed by replacing the elements of $$P.$$ $$A$$ subset $$Q$$ of $$A$$ is again chosen at random. Find the probability that $$P$$ and $$Q$$ have no common elements.

Suppose the probability for A to win a game against B is $$0.4.$$ If $$A$$ has an option of playing either a "best of $$3$$ games" or a "best of $$5$$ games" match against $$B$$, which option should be choose so that the probability of his winning the match is higher ? (No game ends in a draw).