Question

I roll a die repeatedly. Find the chance that the first 4 rolls all show different faces, and the 5th roll shows a face that has appeared before.

Answer 1

The number of combinations of the 6 faces of the die taken 4 at a time is given by 6C4. The number of permutations of the 6 faces of the die taken 4 at a time with repetitions is 6^4.
Therefore the probability that the first 4 rolls app show different faces is given by
\[\frac{6C4}{6^{4}}=\frac{15}{6^{4}}\]
The probability of the fifth roll showing a face that has not appeared before is given by
\[\left(\begin{matrix}4 \\ 0\end{matrix}\right)(\frac{1}{6})^{0}(\frac{5}{6})^{4}=(\frac{5}{6})^{4}\]
Therefore the probability of the fifth roll showing a face that has appeared before is given by
\[1-(\frac{5}{6})^{4}\]
The outcome of the first 4 rolls and the outcome of the fifth roll are independent. Therefore the required probability is given by
\[\frac{15}{6^{4}}\times [1-(\frac{5}{6})^{4}]\]