Let S = { w 1 , w 2 , … } 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} and B = { w n : n A } . Then P ( B ) is equal to [2023]

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Solution

Q.
Let $S = {w_{1}, w_{2}, \dots}$ be the sample space associated to a random experiment. Let $P (w_{n}) = \frac{P (w_{n - 1})}{2}, n \geq 2 .$ Let $A = {2 k + 3 l : k, l \in ℕ}$ and $B = {w_{n} : n \in A}$ . Then $P (B)$ is equal to [2023]

1 $\frac{1}{16}$

2 $\frac{3}{64}$

3 $\frac{3}{32}$

4 $\frac{1}{32}$

Ans.
(2)

Given $S = {w_{1}, w_{2}, \dots}$ be the sample space associated to a random experiment. If

$P (W_{n}) = \frac{P (W_{n - 1})}{2}, n \geq 2$

and let $A = {2 k + 3 l : k, l \in N}$ and $B = {w_{n} : n \in A}$ .

First of all, let $P (w_{1}) = λ$ . Then $P (w_{2}) = \frac{λ}{2}$

$P (w_{n}) = \frac{λ}{2^{n - 1}}$

As
$\sum_{k = 1}^{\infty} P (w_{k}) = 1 \Rightarrow \frac{λ}{1 - \frac{1}{2}} = 1 \Rightarrow λ = \frac{1}{2}$

So $P (W_{n}) = \frac{1}{2^{n}}$

$A = {2 k + 3 l : k, l \in N} = {5, 7, 8, 9, 10, \dots}$

$B = {w_{n} : n \in A}$

$B = {w_{5}, w_{7}, w_{8}, w_{9}, w_{10}, w_{11}, \dots}$

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

$P (B) = 1 - [P (w_{1}) + P (w_{2}) + P (w_{3}) + P (w_{4}) + P (w_{6})]$

$= 1 - [\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{64}] = 1 - \frac{32 + 16 + 8 + 4 + 1}{64} = \frac{3}{64}$

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