Let \[\mathrm{S} = \{ {w_1},{w_2},......\} \] be the sample space associated to a random experiment. Let \[P({w_n}) = {{P({w_{n - 1}})} \over 2},n \ge 2\]. Let \[A = \{ 2k + 3l:k,l \in N\} \] and \[B = \{ {w_n}:n \in A\} \]. Then P(B) is equal to :
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Official solution
\[P({w_1}) + {{P({w_1})} \over 2} + {{P({w_1})} \over {{2^2}}}\, + \,..... = 1\]
\[\therefore\] \[P({w_1}) = {1 \over 2}\]
Hence, \[P({w_n}) = {1 \over {{2^n}}}\]
Every number except 1, 2, 3, 4, 6 is representable in the form
2k+3l where \[k,l \in N\].
\[\therefore\] \[P(B) = 1 - P({w_1}) - P({w_2}) - P({w_3}) - P({w_4}) - P({w_6})\]
\[ = {3 \over {64}}\]