## A Problem of Rolling Six Dice

Problem
Suppose that we roll 6 fair dice (or equivalently, roll a fair die 6 times). Let $X$ be the number of distinct faces that appear. Find the probability function $P(X=k)$ where $k=1,2,3,4,5,6$.

Equivalent Problem
Suppose that we randomly assign 6 candies to 6 children (imagine that each candy is to be thrown at random to the children and is received by one of the children). What is the probability that exactly $k$ children have been given candies, where $k=1,2,3,4,5,6$?

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Discussion
Note that both descriptions are equivalent and are refered to as occupancy problem in [1]. The essential fact here is that $n$ objects are randomly assigned to $m$ cells. The problem then asks: what is the probability that $k$ of the cells are occupied? See the following posts for more detailed discussions of the occupancy problem.

Each of these posts presents different different ways of solving the occupancy problem. The first post uses a counting approach based on the multinomial coefficients. The second post developed a formula for finding the probability that exactly $k$ of the cells are empty.

The first approach of using mulltinomial coefficients is preferred when the number of objects $n$ and the number of cells $m$ are relatively small (such as the problem indicated here). Otherwise, use the formula approach.

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Using the approach of multinomial coefficients as shown in this post (the first post indicated above), we have the following answers:

$\displaystyle P(X=1)=\frac{6}{6^6}=\frac{6}{46656}$

$\displaystyle P(X=2)=\frac{930}{6^6}=\frac{930}{46656}$

$\displaystyle P(X=3)=\frac{10800}{6^6}=\frac{10800}{46656}$

$\displaystyle P(X=4)=\frac{23400}{6^6}=\frac{23400}{46656}$

$\displaystyle P(X=5)=\frac{10800}{6^6}=\frac{10800}{46656}$

$\displaystyle P(X=6)=\frac{720}{6^6}=\frac{720}{46656}$

For more practice problems on calculating the occupancy problem, see this post.
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Reference

1. Feller, W., An Introduction to Probability Theory and its Applications, Vol. I, 3rd ed., John Wiley & Sons, Inc., New York, 1968