# Calculating Expected Value with Combinatorics

A company has 100 employees. For a Christmas team-building exercise, they divide the 100 employees into 25 teams of 4.

Randomly and independently, they give an award to 20 employees.

--

Find the expected value of the number of awards given to the employees on any given team. What type of distribution is this?

So far, I have that there are 100!/(4!^25)*25! combinations for the teams, but don't know where to go from there

Each employee expects $$20/100=1/5$$ of an award, so by linearity of expectation, each team of $$4$$ expects to get $$4/5$$ many awards.

• thank you! I overthought that a little too much. If I wanted to find the expected value of the number of teams with 0 awards, is this correct: (.1)^0 * (.9) ^5 * 20? – alexed Dec 12 '19 at 4:42

We have $$n$$ total people and $$m$$ with awards and the $$n$$ people put into groups of size $$k$$. Since the awards are given uniformly and independently it is sufficient to look at the distribution of awards to first $$k$$ people.

To get $$i$$ award winners in the first $$k$$ - Select $$i$$ people among the $$m$$ award winners - Select $$k - i$$ people among the $$n - m$$ award winners

The number of ways of doing that is

$${m \choose i} \cdot {n - m \choose k - i}$$.

The number of ways of choosing $$k$$ out of $$n$$ people is $${n \choose k}$$.

So the probability of exactly $$i$$ out of $$k$$ people having the award is the ratio.

• The question asks for expectation, not probabilities. – saulspatz Dec 12 '19 at 3:45
• The question asked for how to think about the expectation and distribution. – David Nehme Dec 12 '19 at 4:26