Question

A fair die with faces {1, 2, 3, 4, 5, 6} is thrown repeatedly till '3' is observed for the first time. Let X denote the number of times the die is thrown. The expected value of X is .

• A. 6

Answer 1

The correct option is A 6

Let x = {No. of tosses} = {1, 2, 3, 4, ...}

Probability of getting $3=\frac{1}{6}\text{(}i.e.\left)P\right(W)=\frac{1}{6}$

Probability of not getting $3=1-\frac{1}{6}=\frac{5}{6}i.e.P\left(L\right)=\frac{5}{6}$


So probability distribution in

X:	1	2	3	4	...
P(X)	1/6	$\frac{5}{6},\frac{1}{6}$	${\left(\frac{5}{6}\right)}^2,\frac{1}{6}$		...

If dice thrown repeatedly till first 3 observed first time then $E\left(x\right)=\sum pi{x}_i$

$=\frac{1}{6}+2(\frac{5}{6}.\frac{1}{6})+3(\frac{5}{6}.\frac{5}{6}.\frac{1}{6})+4{\left(\frac{5}{6}\right)}^3.\frac{1}{6}+...$

$=\frac{1}{6}[1+2.\frac{5}{6}+3.{\left(\frac{5}{6}\right)}^2+4{\left(\frac{5}{6}\right)}^3+.....]$

$=\frac{1}{6}{[1-\frac{5}{6}]}^{-2}=\frac{1}{6}{\left[\frac{1}{6}\right]}^{-2}=\frac{36}{6}=6$