The mathematics department has 8 graduate assistants who are assigned to the same office. Each assistant is just likely to study at home as in the office. How many desks must there be in the office so that each assistant has a desk at least 90% of the time?

Let X be the number of a graduate assistants.

⇒ X = 8

Let p be the probability of the assistant studying at the office.

Then, q be the probability of the assistant studying at home.

According to the question, each assistant is just likely to study at home as in the office.

Also, we know that (p + q) = 1.

⇒ q = 1 – p

Now, let there be k number of desks in the office.

We need to find the number of desks in the office so that each assistant has a desk at least 90% of the time.

⇒ P (X > k) < 0.10

Now, we must find a value of k that satisfies the above equation.

Clearly,

P (X > 5) = P (X = 6 or X = 7 or X = 8)

⇒ P (X > 5) = 0.14

Also, P (X > 6) = P (X = 7 or X = 8)

⇒ P (X > 6) = 0.035

∴, P (X > 6) < 0.10

Hence, if there are 6 desks, then there is atleast 90% chance for every graduate assistant to get a desk.

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