A fair die is rolled 6 times, what is the probability that the rolls were exactly 1-6 in sequence?

Question

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.

Answer 1

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 $$\frac{1}{6}\cdot\frac{1}{6}\cdot \frac{1}{6}\cdot\frac{1}{6}\cdot\frac{1}{6}\cdot\frac{1}{6} = \frac{1}{46656}.$$

Answer 2

Hint: Since each sequence of six rolls is equally likely, the answer is $$\frac{1}{\text{number of possible sequences of six rolls}}.$$

Answer 3

On each of the six independent rolls there is a probability of $1/6$ for rolling the desired number, so by the multiplicative principle, the probability for rolling the required sequence is:

$${\left(\dfrac 16\right)}^6~=~\dfrac1{46656}$$