Let $S = \{w_1, w_2, \ldots\}$ be the sample space associated to a random experiment. Let $P(w_n) = \frac{P(w_{n-1})}{2}$, $n \geq 2$. Let $A = \{2k+3\ell; k, \ell \in \mathbb{N}\}$ and $B = \{w_n; n \in A\}$. Then P(B) is equal to
Step-by-Step Solution
Key Concept: Find $P(w_n) = 1/2^n$ from the geometric series, then identify set A = $\mathbb{N} \setminus \{1,2,3,4,6\}$ and compute P(B) by complement
$P(w_1) = 1/2$ (from geometric series sum = 1). $P(w_n) = 1/2^n$. $A = \{5,7,8,9,10,11,...\} = \mathbb{N}\setminus\{1,2,3,4,6\}$. $P(B) = 1 - \sum_{n=1,2,3,4,6} 1/2^n = 1 - (1/2+1/4+1/8+1/16+1/64) = 1-61/64 = 3/64$. Answer: (2)
Correct Answer: $\frac{3}{64}$