Seven cards numbered from $1$ to $7$ are randomly shuffled and then arranged in a circle. Given that the $1$ and $7$ are adjacent to each other, what is the expected number of cards that are larger than both of their neighbors?
I am not sure I understand the question correctly. But here's my thought process:
If $1$ and $7$ are adjacent, their neighbors can be chosen in $5C_2 = 10$ ways, therefore,if the neighbors are $2$ and $3$, for instance, these can come in $2$ ways $(2,3)$ and $(3,2)$, hence it's probability is $\frac{2}{10}$ and the numbers larger than $2$ and $3$ are $4,5,$ and $6$. Working on individual cases like this I arrived at the following:
$\frac{2}{10} \times 3 + \frac{2}{10} \times 2 + \frac{2}{10} \times 1 + \frac{2}{10} \times 2 + \frac{2}{10} \times 1 + \frac{2}{10} \times 1 = 2$
Therefore, the answer is $2$, but this is wrong. I simulated it and I got $\approx 2.5$. How can I solve this question?