Let rn be the total number of non-overlapping success runs

Question

Success runs in independent trials. Consider a sequence of n > 4 independent trials say  run of three successes occurs at the beginning of the sequence if the first four trials result in SSF: and a run of three successes elsewhere in the sequence is the pattern FSSS: and a run of three successes elsewhere in the sequence is the pattern FSSF.  Let R₃,_ndenote the number of runs of three successes in the n trials

a) Find E(R3,_n).

b) Define R_m,n, the number of success runs of length m in n trials, similarly for 1 < m="">< n.="" find="">_m,n).

c) Let R_n be the total number of non-overlapping success runs in n trials, counting runs of any length between 1 and n. Find E(R_n) by using the result of b).

d) Find E(R_n) another way by considering for each 1 j n the number of runs that start on the jth trial. Check that the two methods give the same answer.