I am studying for an exam and found the following problem.
Let $L = l_0, l_1, ... , l_{n+1}$ be a list of items. For each item from 0 to $n+1$, we flip a coin (fairly). We add item $l_i$ (with $1 \le i \le n$) to a set if $l_i$'s toss came up heads, and both of its neighbours ($l_{i - 1}$ and $l_{i + 1}$) came up tails. What is the probability that $l_i$ is included in the set?
My initial thought is that it is simply 1/8. The probability that the coin is heads is 1/2, the one before it being tails has probability 1/2, and the same idea for the toss after it. Am I missing something, or is it really that simple? They are obviously dependent on one another.