A fair coin in tossed until a head or five tails occur. If X denotes the number of tosses of the coin, then mean of X is 

$S=\{H,TH,TTH,TTTH,TTTTH,TTTTT\}$

$X=1,2,3,4,5$

$P(X=1)=P\left(H\right)=\frac{1}{2}$

$P(X=2)=P\left(TH\right)=\frac{1}{2}\left(\frac{1}{2}\right)=\frac{1}{2^2}=\frac{1}{4}$

$P(X=3)=\frac{1}{2^3}=\frac{1}{8}=P\left(TTH\right)$

$P(X=4)=\frac{1}{24}=\frac{1}{16}=P\left(TTTH\right)$

$P(X=5)=P\left(TTTTH\right)+P\left(TTTTT\right)=\frac{1}{2^5}+\frac{1}{2^5}=\frac{2}{2^5}=\frac{1}{16}$

$\begin{array}{l}E\left(x\right)=\sum x.P(X=x)\\ =1\left(\frac{1}{2}\right)+2\left(\frac{1}{4}\right)+3\left(\frac{1}{8}\right)+4\left(\frac{1}{16}\right)+5\left(\frac{1}{16}\right)\\ =\frac{31}{16}=1.9\end{array}$