Our next calculation will be for 5 spins of a three-numbered roulette wheel.

Using our previous method, this would require listing 243 five digit numbers and then choosing how many of those contain 1, 2 and 3.
At this point, you can see that using this “brute-force” method is very laborious and the possibility of making a mistake is extremely high.

Now, the calulations will be more mathemaically-oriented.

STEP 1
First, we calculate the value of “n” (numbers on the roulette wheel) raised to the power of “r” (number of spins).

3⁵ = 243This represents all possible outcomes when spinning a three-numbered roulette wheel five times.

STEP 2
Next, we calculate each value of “n” raised to the power of “r”.
In this case “n” = 3 numbers on the roulette wheel and r = “5” spins.
(And, yes we’ve already calculated 3⁵, so now we will calculate 2⁵ and 1⁵.)

3⁵ = 243
2⁵ = 32
1⁵ = 1STEP 3
Next, we calculate how many combinations can be made from “n” objects for each value of “n”.

3 C 3 = 1
2 C 3 = 3
1 C 3 = 3          Basically, this is saying that

3 objects can be chosen from a set of 3 in 1 way
2 objects can be chosen from a set of 3 in 3 ways
1 object can be chosen from a set of 3 in 3 waysSTEP 4
We then calculate the product of the first calculation of STEP 2 times the first calculation of STEP 3 and do so throughout all 3 numbers.

3⁵ • 1 = 243
2⁵ • 3 =   96
1⁵ • 3 =   3STEP 5
Then, alternating from plus to minus, we sum the 3 numbers we just calculated.

+ 243
– 96
+ 3TOTAL 150

which equals the total number of ways you can spin a three digit roulette wheel 5 times and have all 3 numbers appear.STEP 6
So, if we take the number 150 and divide it by
243 (all posiible results of 5 spins of a 3 numbered roulette wheel)
we get the probability of having all 3 numbers appearing after 5 spins.

150 ÷ 243 = 0.617283950617284So, if we had a roulette wheel with 3 numbers on it, we would need to spin it 5 times in order to have a 50% (or better) chance to get all 3 numbers to appear.

In the next 5 pages, we have calculated the occupancy probabilities of rolling dice that have 4, 6, 8, 12 and 20 sides.
(If you are wondering, these would be dice that are in the shape of the 5 Platonic Solids.)
(Tetrahedron, Hexahedron, Octahedron, Dodecahedron, Icosahedron)

Click here to see the probabilities of a:

4 Sided Die  Probability of all 4 numbers in 7 Rolls

6 sided die  Probability of all 6 numbers in 13 Rolls

8 Sided Die  Probability of all 8 numbers in 20 Rolls

12 Sided Die  Probability of all 12 numbers in 35 Rolls

20 Sided Die  Probability of all 20 numbers in 67 Rolls