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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.

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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

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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.

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  • $\begingroup$ 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? $\endgroup$ – alexed Dec 12 '19 at 4:42
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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.

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

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