Question

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

Answer 1

Let k be the number of desks and X be the number of graduate assistants in the office.
therefore, X=8, \[p = \frac{1}{2}, q = \frac{1}{2}\]

According to the given condition, 

\[P\left( X \leq k \right) > 90\] % 
\[ \Rightarrow P\left( X \leq k \right) > 0 . 90\]
\[ \Rightarrow P\left( X > k \right) < 0 . 10\]
\[ \Rightarrow P(X = k + 1, k + 2, . . . . 8) < 0 . 10\]

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

\[^{8}{}{C}_7 \left( \frac{1}{2} \right)^8 + ^{8}{}{C}_8 \left( \frac{1}{2} \right)^8 = 0 . 04\]

Now, P(X > 5) = P(X = 6, X = 7 or X = 8) = 0.15
P(X > 6)  < 0.10
        
So, if there are 6 desks then there is at least 90% chance for every graduate to get a desk.