Occupancy Probability 6 Sided Die
(Hexahedron)

For this problem, we'll calculate the probabilty of getting all 6 numbers after rolling a 6-sided die 13 times.

STEP 1
The set of numbers on the die (1 through 6), would be called called "n" and the number of trials or attempts is called "r" which in this case is 13 rolls.
Calculating the value of "n" raised to the power of "r":

613
which equals
13,060,694,016

This number represents all the possible results from rolling a 6 sided die

In steps 2, 3, 4 and 5, we will determine how many of those 613 rolls, will contain all 6 numbers.

STEP 2
We first must calculate each value of "n" raised to the power of "r".
Rather than explain, this is much easier to show:

613 = 13,060,694,016
513 = 1,220,703,125
413 = 67,108,864
313 = 1,594,323
213 = 8,192
113 = 1

STEP 3
Next, we calculate how many combinations can be made from "n" objects for each value of "n".
This is much easier to show than explain:

6 C 6 = 1
5 C 6 = 6
4 C 6 = 15
3 C 6 = 20
2 C 6 = 15
1 C 6 = 6

Basically, this is saying that
6 objects can be chosen from a set of 6 in 1 way
5 objects can be chosen from a set of 6 in 6 ways
4 objects can be chosen from a set of 6 in 15 ways
3 objects can be chosen from a set of 6 in 20 ways
2 objects can be chosen from a set of 6 in 15 ways
1 object can be chosen from a set of 6 in 6 ways

STEP 4
We then calculate the product of the first number of STEP 2 times the first number of STEP 3 and do so throughout all 6 numbers.

13,060,694,016     ×   1     = 13,060,694,016
1,220,703,125   ×   6     = 7,324,218,750
67,108,864     ×   15     = 1,006,632,960
1,594,323     ×   20   = 31,886,460
8,192   × 15     = 122,880
1     ×   6     = 6

STEP 5
Then, alternating from plus to minus, we sum the 6 terms we just calculated.

 + 13,060,694,016 - 7,324,218,750 + 1,006,632,960 - 31,886,460 + 122,880 -6 Total 6,711,344,640

which equals the total number of ways you can roll a 6 sided die 13 times and have all 6 numbers appear.

STEP 6
So, if we take the number 6,711,344,640 and divide it by
13,060,694,016 (all posiible results of 13 rolls of a 6 sided die)
we get the probability of having all 6 numbers appearing after 13 rolls.

6,711,344,640 ÷ 13,060,694,016 = 0.513858194042236

So, if you had a 6 sided die, you would need to roll it at least 13 times in order to have a better than 50 / 50 chance of rolling all 6 numbers.