A Tourist Survey

Visitors to a landmark are surveyed one at a time about whether they plan to visit a second attraction. A visitor who declines scores $1$ point and a visitor who accepts scores $2$ points; each visitor accepts or declines with probability $\dfrac{1}{2}$, independently of the others.

1. Three visitors are chosen at random; let $X$ be their total score. Find the distribution and the expected value of $X$. 2. Answer the following.

1. If $m$ visitors are chosen at random, let $A_m$ be the probability that their total score is exactly $m$ points. Find the sum of the first $10$ terms of the sequence $\{A_m\}$.
2. As visitors are surveyed one after another, let $B_n$ be the probability that the running total is at some point exactly $n$. Find a relation between $B_n$ and $B_{n-1}$, and a closed formula for $B_n$.