permutations rolling die 6 times

Question

How would you calculate the probability that if you roll a six sided die six times you will roll 1,2,3,4,5,6 consecutively. I am totally lost on how to even calculate this.

Answer 1

How many possible sequences of $6$ rolls of a six-sided die are there? How many of those sequences meet the desired criterion?

Now, how is the probability of an event defined, in general? For example, if you are choosing a shirt from your closet in the pitch dark, and you have $5$ black shirts out of a total of $18$ shirts, then what is the probability that the shirt you pick is black?

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Another approach: $\frac16$ of the time, the first roll should be $1.$ $\frac16$ of that time, the second roll should be $2$. $\frac16$ of that time, the third roll should be $3,$ and so on. Hence, $\frac16$ of $\frac16$ of $\frac16$ of $\frac16$ of $\frac16$ of $\frac16$ of the time, the sequence of rolls should be $1,2,3,4,5,6.$ How can we express the number "$\frac16$ of $\frac16$ of ..." more simply?

Answer 2

The probability of obtaining, specifically and in this order: $1, 2, 3, 4, 5, 6$ is the same probability of obtaining any particularly specified sequence.

There are $6$ possible numbers for filling each position. And the probability of obtaining any particular specified number in one particular position is $\dfrac 16$. There are six rolls, therefor six different positions in the sequence. So we have a probability of $$\underbrace{\dfrac 16 \cdot \frac 16 \cdot \cdots \dfrac 16}_{6\;\text{ factors}} = \left(\dfrac 16\right)^6$$