Question

The expected number of failures preceding the first success in an infinite series of independent trials with constant probability $p$ is

Answer 1

The probability of success in $1^{st},2^{nd},\&3^{rd}$ trials respectively are $p,qp,q^{2}p,q^{3}p,......$

$$\begin{gathered} E\left(x\right)=\left(0.p\right)+\left(1.qp\right)+\left(2.q^{2}p\right)+..... \\ =qp\left(1+2q+3q^2\hspace{0.17em}+...\right)...\left(Whereq<1\right) \\ =\frac{qp}{{\left(1-q\right)}^2} \\ =\frac{qp}{p^2} \\ =\frac{q}{p} \\ =\frac{1-p}{p}...\left(P+q=1\right) \end{gathered}$$

Hence, the expected number of failures preceding the first success in an infinite series of independent trials with constant probability $p$ is $\frac{1-p}{p}$