Question

A coin is tossed repeatedly until it shows head. Let $X$ be the number of tosses required to get head. Write the probability distribution of $X$.

Answer 1

Suppose the head appears in $x^{th}$ toss. This means that tail appears in all preceding $(x-1)$ tosses.

Now, if p is the probability of getting a head then probability of getting a tail is $1-p=q$. So, the probability function of x will be $f\left(x\right)=q^{x-1}p$ for $x=1,2,3$….. Now the expected of tosses will be

$E\left(x\right)={\epsilon }_{x=1}^{\infty }x{q}^{x-1}p=\frac{p}{q}{\epsilon }_{x=1}^{\infty }x{q}^x=\frac{p}{q}q(1+2q+3q^2+....)$

$=\frac{p}{q}q(1-q{)}^{-2}=\frac{1}{pq}$.