A fair die with faces $$\left\{ {1,2,3,4,5,6} \right\}$$ is thrown repeatedly till $$'3'$$ is observed for the first time. Let $$X$$ denote the number of times the dice is thrown. The expected value of $$X$$ is _________.

The input $$X$$ to the Binary Symmetric Channel (BSC) shown in the figure is $$'1'$$ with probability $$0.8.$$ The cross-over probability is $$1/7$$. If the received bit $$Y=0,$$ the conditional probability that $$'1'$$ was transmitted is _______.

Parcels from sender $$S$$ to receiver $$R$$ pass sequentially through two post - offices. Each post - office has a probability $${1 \over 5}$$ of losing an incoming parcel, independently of all other parcels. Given that a parcel is lost, the probability that it was lost by the second post - office is _________.

A fair coin is tossed repeatedly till both head and tail appear at least once. The average number of tosses required is ________.