Consider a die with the property that the probability of a face with $$'n'$$ dots showing up is proportional to $$'n'.$$ The probability of the face with three dots showing up is _________.

Let $$X$$ be a random variable with probability density function $$f\left( x \right) = \left\{ {\matrix{ {0.2} & {for\,\left| x \right| \le 1} \cr {0.1} & {for\,1 < \left| x \right| \le 4} \cr 0 & {otherwise} \cr } } \right.$$

The probability $$P\left( {0.5 < x < 5} \right)$$ is _________.

A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, is

A fair coin is tossed $$3$$ times in succession. If the first toss produces a head, then the probability of getting exactly two heads in three tosses is