Let S = \ w _ 1 , w _ 2 , .\ be the sample space associated to a...

Given,

$P\left({w}_{n}\right) = \dfrac{P\left({w}_{n - 1}\right)}{2}, n \ge 2$

Let $P\left({w}_{1}\right) = \lambda$ then

$P\left({w}_{2}\right) = \dfrac{\lambda}{2}$

$P\left({w}_{3}\right) = \dfrac{\lambda}{{2}^{2}}$

$⋮ ⋮ ⋮ ⋮ ⋮ ⋮$

$P\left({w}_{n}\right) = \dfrac{\lambda}{{2}^{n - 1}}$

As

$\mathop{∑}_{k = 1}^{\infty}P\left({w}_{k}\right) = 1$

$⇒ \lambda + \dfrac{\lambda}{2} + \dfrac{\lambda}{{2}^{2}} + \dfrac{\lambda}{{2}^{3}} + . . . . = 1$

$⇒ \dfrac{\lambda}{1 - \dfrac{1}{2}} = 1 ⇒ \lambda = \dfrac{1}{2}$

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

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

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

$B = \left\{{w}_{5}, {w}_{7}, {w}_{8}, {w}_{9}, {w}_{10}, {w}_{11}, \ldots\right\}$

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

$∴ P\left(B\right) = 1 - \left[P\left({w}_{1}\right) + P\left({w}_{2}\right) + P\left({w}_{3}\right) + P\left({w}_{4}\right) + P\left({w}_{6}\right)\right]$

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

$= 1 - \left(\dfrac{32 + 16 + 8 + 4 + 1}{64}\right) = \dfrac{3}{64}$