1. Tardigrade
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  3. Mathematics
  4. Let S= w1, w2, ldots. be the sample space associated to a random experiment. Let P ( w n )=( P ( w n -1)/2), n ≥ 2. Let A = 2 k +3 ℓ ; k , ℓ ∈ N and B = w n ; n ∈ A . Then P ( B ) is equal to

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Q. 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

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Solution:

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 $\displaystyle\sum_{ k =1}^{\infty} P \left( w _{ k }\right)=1 \Rightarrow \frac{\lambda}{1-\frac{1}{2}}=1 \Rightarrow \lambda=\frac{1}{2}$
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] $
$=1-\frac{32+16+8+4+1}{64}=\frac{3}{64}$