GATE Electrical Engineering (EE) 2021 Solved Papers

Question : 44 of 65

Marks: +1, -0

Suppose the probability that a coin toss shows "head" is

{p}

, where

0< {p}<1

. The coin is tossed repeatedly until the first "head" appears. The expected number of tosses required is:

Solution:

Let X denotes the number of tosses required to get the first head,

X	1	2	3	4	........
P(X)	p	pq	$\pq^2$	$\pq^3$	........

E(x)=∑ x ⋅ P(x)

{E}( {x})= {p}+2 {pq}+3 {pq}^{2}+4 {pq}^{3}+...

{E}( {x})= {p}[1+2 {q}+3 {q}^{2}+4 {q}^{3}+.....]

∵ 1+2 {A}+3 {A}^{2}+4 {A}^{3}+. . .+(1- {A})^{-2}

{E}( {x})= {p}[1- {q}]^{-2} ∵ {p}+ {q}=1

{E}( {x})= {p} ⋅ {p}-2 {p}=1- {q}

E(x)=\;{1}/{p}

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