https://www.bigocheatsheet.com considering finding (Access) the position of the element before insert as separate operation.
Access - O(1) // we can get the element by index directly
Insertion - O(n) // in the worst case we need to resize the array to have a space for the new element
Access - O(n) // we need to reach the element
The probability $1/i$ is correct, since it refers to the relative order of $a_1,\ldots,a_i$ before sorting the first $i-1$ elements.
However, the argument seems wrong. The relevant probability is not $p_i(k)$, but rather $q_i(k)$, which is the probability that $a_i$ is the $k$'th smallest element out of $a_1,\ldots,a_i$ (before sorting). This probability is ...
Let us assume that we are given a real number $x$ and a list of $n$ distinct real numbers. The linear search algorithm, described below,
while(i<=n and x not equal to ai):
locates $x$ by successively ...