Exercise 1.36

Question

A group of 30 dice are thrown. What is the probability that 5 of each of the values 1, 2, 3, 4, 5, 6 appear?

fifthist x · Accepted Answer

We can think of the problem as sampling with replacement where order matters.

There are $6^{30}$ possible sequences of outcomes. We are interested in the cases 
where each face of the die is rolled exactly $5$ times. Since each sequence is 
equally likely, we can use the naive definition of probability.

There are $\binom{30}{5}$ ways to select the dice that fall on a $1$. Then, 
$\binom{25}{5}$ ways to select the dice falling on a $2$, $\binom{20}{5}$ 
falling on a $3$, $\binom{15}{5}$ falling on a $4$, $\binom{10}{5}$ falling 
on a $5$ and finally, $\binom{5}{5}$ falling on a $6$.

Thus, the desired probability is $$\frac{\binom{30}{5} \binom{25}{5} 
\binom{20}{5} \binom{15}{5} \binom{10}{5} \binom{5}{5}}{6^{30}}$$

Exercise 1.36

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Exercise 1.36

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