Question

Jeeves is a butler with 6 bathrooms. He's too busy to clean every bathroom everyday. Instead he randomly chooses a bathroom everyday and cleans it.
1) what is the probability that after 30 days, exactly 1 bathroom has never been cleaned?
2) FUZZY THE CAT COUGHS UP A hairball in the master bedrppm. How long should we expect to wait until jeeves cleans it up?

Answer 1

probability of non clean for each bathroom in each day is 5/6
what is the probability of 1 not clean in thirty trials?
Probability of non clean = 5/6
number of trials 30
Bernoulli (n/k) is binomial coef
p k non cleans in n trials = (n/k)(5/6)^k(1-5/6)^(n-k)
here k = 1 and n=30
(30/1) = 30!/[29!1!] = 30
so probability of exactly one non clean in 30 trials is
30 (5/6)^6 (1/6)^24

Answer 2

for the second part it is a particular bathroom of the 6
The probability that this bathroom is not cleaned each day is 5/6
after n days the probability is (5/6)^n
how many days until the probability is like 1/2 ?
1/2 = (5/6)^n
log .5 = n log (5/6)
-.301 = n (-.07918)
n = 3.8 days
so the chances are 50/50 that Jeevs got it by the fourth day