A box contains $$N$$ coins, $$m$$ of which are fair and the rest are biased. The probability of getting a head when a fair coin is tossed is $$1/2$$, while it is $$2/3$$ when a biased coin is tossed. A coin is drawn from the box at random and is tossed twice. The first time it shows head and the second time it shows tail. what is the probability that the coin drawn is fair?

An urn contains $$m$$ white and $$n$$ black balls. A ball is drawn at random and is put back into the urn along with $$k$$ additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. What is the probability that the ball drawn now is white?

An unbiased die, with faces numbered $$1,2,3,4,5,6,$$ is thrown $$n$$ times and the list of $$n$$ numbers showing up is noted. What is the probability that, among the numbers $$1,2,3,4,5,6,$$ only three numbers appear in this list?

A coin has probability $$p$$ of showing head when tossed. It is tossed $$n$$ times. Let $${p_n}$$ denote the probability that no two (or more) consecutive heads occur. Prove that $${p_1} = 1,{p_2} = 1 - {p^2}$$ and $${p_n} = \left( {1 - p} \right).\,\,{p_{n - 1}} + p\left( {1 - p} \right){p_{n - 2}}$$ for all $$n \ge 3.$$