A fair die is rolled 6 times, what is the probability that the rolls were exactly 1-6 in sequence?
Thanks to an anime I'm watching I'm suddenly curious about this. A few similar questions have given me some input, but as it's been a very long time since I've battled with probability questions I'll likely reach an incorrect solution.
Best Answer
This question is equivalent to, what is the probability any particular sequence will appear if a dice is rolled $6$ times, the fact that this particular sequence happens to be $1,2,3,4,5,6$ is irrelevant. Hence there is a $\frac{1}{6}$ chance a $1$ will be rolled first, $\frac{1}{6}$ chance a $2$ will be rolled second, $\frac{1}{6}$ that a $3$ will be rolled third, etc. Therefore the probability of the sequence appearing is $$\frac{1}{6}\cdot\frac{1}{6}\cdot \frac{1}{6}\cdot\frac{1}{6}\cdot\frac{1}{6}\cdot\frac{1}{6} = \frac{1}{46656}.$$