Data management

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?

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  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

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  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

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