An information source generates symbols at random from a four$-$letter alphabet $,$

$d$ with probabilities $P(a)=0\mathrm{.}5P(b)=0\mathrm{.}25,$ and $P(c)=P(d)=0.125.$ A coding scheme

encodes these symbols into binary codes as follows:

Let X be the r$.$v$.$ denoting the length of the code$-$that is$,$ the number of binary symbols

$($bits$).$ Assuming that the generations of symbols are independent, find the probabilities

$P(x=1),P(x=2),P(x=3),$ and $P(x>3).$