Question

3. (10 points) A fair, 6-sided die is rolled continually until two consecutive 6s appear. Find the expected number of rolls. (Hint: Ross Chapter 3, Exercise Q23)

Answer 1

x=1 sol =) The expected no. of rolls required is - defined as . ECX) = 2 x.P[x= x] To get first six , we required on an average 6 solls. suppose we get again 6 in next draw with probablity Yo and get another number from I to s with probablity sl. so if we do not get 6 in one more roll after getting first 8 6 in on an average o draw then get it again affee on an average 7, rolls .. so now we find EX) = 3 ***(+) Elx) = X. (E)* €(X) = x 2 (E) e(x) as (+ 2. (=]+($]+ -] s (1-51632 = x36 le (x) = 6) more . Elx) z ELX) = El-sios e (x) = 6

to get The expected no. of rolls required two consecutive 6's is E = 6+ Ļx1 + EXLE+ E - SE = 6+to+ = = 6+1 6 + ŽE E = 7x6 [E=42 E242