## Success runs in Independent trials

Consider a sequence of n>=4 indpt trials, each resulting in a success S with prob p and failure F with prob 1-p.
A run of 3 consecutive successes occurs at beginning of the sequence as SSSF, at the end of the sequence as FSSS or in the middle as FSSSF. Let R3,n denote the number of runs of three successes in n trials.

a) Find E(R3,n)
b) For 1<=m<=n, define Rm,n similarly and find E(Rm,n)
c)If Rn is the total number of non-overlapping runs of successes in n trials, counting runs of any length from 1 to n, find E(Rn)