Answer:
0.9n + 0.1, with n the length of the list.
Step-by-step explanation:
Lets denote with n the length of the list.
80% of the time x is not in the list, so 80% of the time we make n comparisons (or we make n comparisons with probability 0.8).
20% of the time (probability 0.2) we use linear search to find x, since x is equally likely to be at any place on the list, then we should expect that, provivded that x is in the list, we should make (1+2+3+4+....+n)/n comparisons (on average). Using Gauss's sum we conclude that the number is equal to (n+1)*n/(2n) = (n+1)/2 (here we are assuming that x appears only once on the list).
As a conclussion, the average number of comparisons we make is
0.8*n + 0.2*(n+1)/2 = 0.8n + 0.1n + 0.1 = 0.9n + 0.1
I hope that works for you!
