discrete distributions?

Question

How to solve the question and is this poisson.

The average no. of students attended to by teachers is 2.5. How many tutors are needed so that there is not more than 10% chance of a student having to wait.

Answer 1

You need to specify a probability distribution for the number of incoming students. That would most likely be modeled by a Poisson distributed random variable (as quite often in Queuing Theory) with rate lambda>0. I will denote it by X and the number of teachers by n..

Then the criterion for the number of teachers reads as follows:

P(X>n*2.5)<=0.1
1-P(X<=n*2.5)<=0.9
P(X<=n*2.5)>=0.1.
F(n*2.5)>=0.1.

(where F is the cdf of the Poisson distribution)

You can solve it for n, or, for large n, you can also use a normal approximation by using the central limit theorem. Tell me if you're interested how this is done by using "Add details".

Answer 2

let there are n students

1 teacher attends 2.5 students
so, 1 student is attended by 1/2.5 teacher
and, n students are attended by n/2.5 teacher

if there are p students and n/2.5 teachers then the number of students that have to wait is p-n

so p-n = 10% of p = 0.1p
or, 0.9p = n

so for p students there must be 0.9p / 2.5 = 0.36p teachers

if p = 100, then we need 36 teachers
36 teachers serve 36*2.5 = 90 students, and 100-90 = 10 students wait