Let S= w _1, w _2, . be the sample space associated to a random e… - Vidyadip

MCQ

Let $S=\left\{ w _1, w _2, \ldots.\right\}$ be the sample space associated to a random experiment. Let $P \left( w _{ n }\right)=\frac{ P \left( w _{ n -1}\right)}{2}, n \geq 2$.Let $A=\{2 k +3 \ell ; k , \ell \in N \}$ and $B=\left\{ W _{ n } ; n \in A \right\}$.Then $P ( B )$ is equal to

A
$\frac{3}{32}$
B
$\frac{3}{64}$
C
$\frac{1}{16}$
D
$\frac{1}{32}$

✓

Answer

Let $P \left( w _1\right)=\lambda$ then $P \left( w _2\right)=\frac{\lambda}{2} \ldots P \left( w _{ n }\right)=\frac{\lambda}{2^{ n -1}}$

As $\sum \limits_{ k =1}^{\infty} P \left( w _{ k }\right)=1 \Rightarrow \frac{\lambda}{1-\frac{1}{2}}=1 \Rightarrow \lambda=\frac{1}{2}$

So, $P \left( w _{ n }\right)=\frac{1}{2^{ n }}$

$A =\{2 k +3 \ell ; k , \ell \in N \}=\{5,7,8,9,10 \ldots .\}$

$B =\left\{ w _{ n }: n \in A \right\}$

$B =\left\{ w _5, w _7, w _8, w _9, w _{10}, w _{11}, \ldots .\right\}$

$A = N -\{1,2,3,4,6\}$

$\therefore P ( B )=1-\left[ P \left( w _1\right)+ P \left( w _2\right)+ P \left( w _3\right)+ P \left( w _4\right)+ P \left( w _6\right)\right]$

$=1-\left[\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{64}\right]$

$=1-\frac{32+16+8+4+1}{64}=\frac{3}{64}$

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