Question

Problem 8. A clock tower at the Texas A&M campus rings bell every hour.
Every hour, it will ring once with probability 1/3 and twice with probability
2/3. The numbers of times the bell rings at different hours are independent.
If stay on campus for 4 hours (we hear bell on 4 occasions) what is the
probability that we hear in total 7 rings.

Answer 1

Solution:
Given in the question:
Every hour probability that it will ring once = (1/3)
While every hour probabilty that it will ring twice a hour = 2/3
So if in university it stay for 4 hours than probability that it rings for 7 hours(if first hour 1 bell and next three hour is 2 bells) = (1/3)*(2/3)*(2/3)*(2/3) = 0.098765

if first hour 2 bell,2nd hour 1 bell and next two hour is 2 bells) = (2/3)*(1/3)*(2/3)*(2/3) = 0.098765
if first 2 hour 2 bell,3rd hour 1 bell and 4th hour is 2 bells) = (2/3)*(2/3)*(1/3)*(2/3) = 0.098765
If first 3 hours 2 bell, fourth hour is 1 bells = (2/3)*(2/3)*(2/3)*(1/3) = 0.098765
So there is (0.098756*4) = 0.3950 or 39.50% probability that we hear in total 7 rings.