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?
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
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
To answer these questions, we need to understand some probability concepts and calculations.
1) Probability that exactly 1 bathroom has never been cleaned after 30 days:
The probability of Jeeves randomly selecting a specific bathroom on any given day is 1 out of 6, or 1/6. Therefore, the probability of Jeeves not selecting a specific bathroom on a given day is 5 out of 6, or 5/6.
To find the probability that, after 30 days, exactly 1 bathroom has never been cleaned, we need to account for the specific bathroom that hasn't been cleaned and the 29 days that the other 5 bathrooms are cleaned.
The probability of a specific bathroom not being selected on any given day for 30 days is (5/6) * (5/6) * ... * (5/6) (30 times). This can also be written as (5/6)^30.
However, we have 6 different bathrooms that could potentially be the one never cleaned. Therefore, we need to multiply the above probability by 6 to account for all possibilities.
So, the probability can be calculated as: 6 * (5/6)^30 = 0.144 = 14.4%
Therefore, there is a 14.4% probability that, after 30 days, exactly 1 bathroom has never been cleaned.
2) Expected waiting time until Jeeves cleans up the hairball:
Since Jeeves randomly selects one bathroom to clean each day, we can view this as a problem of waiting time until a specific event (cleaning the master bedroom) occurs.
The waiting time until a specific event (cleaning the master bedroom) happens follows a geometric distribution. The expected waiting time for a geometric distribution can be calculated as the reciprocal of the probability of the event occurring on any given day.
In this case, the probability of cleaning the master bedroom on any given day is 1 out of 6, or 1/6.
So, the expected waiting time can be calculated as: 1 / (1/6) = 6 days.
Therefore, we can expect to wait approximately 6 days until Jeeves cleans up the hairball in the master bedroom.