Occupancy Probability for 38 Number Roulette Wheel

For our next problem, let us calculate the probabilty of getting all 38 numbers after spinning a roulette wheel 152 times.

(An American roulette wheel has the numbers 1 through 36 plus zero and double zero.)

STEP 1
To determine the number of all the results of spinning a 38 number roulette wheel 152 times we raise "n" to the power of "r":

38152
which equals
1.34 × 10240

Then in steps 2, 3, 4 and 5, we will determine how many of those 38152 spins, will contain all 38 numbers.

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

38152 = 1.34 × 10240
37152 = 2.33 × 10238
36152 = 3.61 × 10236
35152 = 4.99 × 10234
34152 = 6.09 × 10232
33152 = 6.52 × 10230
32152 = 6.06 × 10228
31152 = 4.86 × 10226
30152 = 3.33 × 10224
29152 = 1.93 × 10222
28152 = 9.29 × 10219
27152 = 3.69 × 10217
26152 = 1.19 × 10215
25152 = 3.07 × 10212
24152 = 6.20 × 10209
23152 = 9.61 × 10206
22152 = 1.12 × 10204
21152 = 9.49 × 10200
20152 = 5.71 × 10197
19152 = 2.35 × 10194
18152 = 6.33 × 10190
17152 = 1.07 × 10187
16152 = 1.06 × 10183
15152 = 5.83 × 10178
14152 = 1.63 × 10174
13152 = 2.09 × 10169
12152 = 1.09 × 10164
11152 = 1.96 × 10158
10152 = 1.00 × 10152
9152 = 1.11 × 10145
8152 = 1.86 × 10137
7152 = 2.85 × 10128
6152 = 1.90 × 10118
5152 = 1.75 × 10106
4152 = 3.26 × 1091
3152 = 3.33 × 1072
2152 = 5.71 × 1045
1152   =     1

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

38 C 38 = 1
37 C 38 = 37
36 C 38 = 703
35 C 38 = 8,436
34 C 38 = 73,815
33 C 38 = 501,942
32 C 38 = 2,760,681
31 C 38 = 12,620,256
30 C 38 = 48,903,492
29 C 38 = 163,011,640
28 C 38 = 472,733,756
27 C 38 = 1,203,322,288
26 C 38 = 2,707,475,148
25 C 38 = 5,414,950,296
24 C 38 = 9,669,554,100
23 C 38 = 15,471,286,560
22 C 38 = 22,239,974,430
21 C 38 = 28,781,143,380
20 C 38 = 33,578,000,610
19 C 38 = 35,345,263,800
18 C 38 = 33,578,000,610
17 C 38 = 28,781,143,380
16 C 38 = 22,239,974,430
15 C 38 = 15,471,286,560
14 C 38 = 9,669,554,100
13 C 38 = 5,414,950,296
12 C 38 = 2,707,475,148
11 C 38 = 1,203,322,288
10 C 38 = 472,733,756
9 C 38 = 16,3011,640
8 C 38 = 48,903,492
7 C 38 = 12,620,256
6 C 38 = 2,760,681
5 C 38 = 501,942
4 C 38 = 73,815
3 C 38 = 8,436
2 C 38 = 703
1 C 38 = 38

Basically, this is saying that
38 objects can be chosen from a set of 38 in 1 way
37 objects can be chosen from a set of 38 in 37 ways
36 objects can be chosen from a set of 38 in 703 ways
.......................................................................................

2 objects can be chosen from a set of 38 in 703 ways
1 object can be chosen from a set of 38 in 38 ways

STEP 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 38 numbers.

For example,
1.34 × 10240 × 1 = 1.34 × 10240
2.33 × 10238 × 37 = 8.84 × 10239
and so on
 1.34 × 10240 8.84 × 10239 2.54 × 10239 4.21 × 10238 4.50 × 10237 3.27 × 10236 1.67 × 10235 6.14 × 10233 1.63 × 10232 3.14 × 10230 4.39 × 10228 4.44 × 10226 3.22 × 10224 1.66 × 10222 5.99 × 10219 1.49 × 10217 2.49 × 10214 2.73 × 10211 1.92 × 10208 8.30 × 10204 2.13 × 10201 3.07 × 10197 2.36 × 10193 9.02 × 10188 1.57 × 10184 1.13 × 10179 2.94 × 10173 2.36 × 10167 4.73 × 10160 1.81 × 10153 9.1 × 10144 3.60 × 10135 5.25 × 10124 8.79 × 10111 2.41 × 1096 2.81 × 1076 4.01 × 1048 38 Total   1.36 × 10240

STEP 5
Then, alternating from plus to minus, we sum the 38 terms we just calculated.

 + 1.34 × 10240 - 8.84 × 10239 + 2.54 × 10239 - 4.21 × 10238 + 4.50 × 10237 - 3.27 × 10236 + 1.67 × 10235 - 6.14 × 10233 + 1.63 × 10232 - 3.14 × 10230 + 4.39 × 10228 - 4.44 × 10226 + 3.22 × 10224 - 1.66 × 10222 + 5.99 × 10219 - 1.49 × 10217 + 2.49 × 10214 - 2.73 × 10211 + 1.92 × 10208 - 8.30 × 10204 + 2.13 × 10201 - 3.07 × 10197 + 2.36 × 10193 - 9.02 × 10188 + 1.57 × 10184 - 1.13 × 10179 + 2.94 × 10173 - 2.36 × 10167 + 4.73 × 10160 - 1.81 × 10153 + 9.10 × 10144 - 3.60 × 10135 + 5.25 × 10124 - 8.79 × 10111 + 2.41 × 1096 - 2.81 × 1076 + 4.01 × 1048 - 38 Total   + 6.72 × 10239

6.72 × 10239 equals the total number of ways a 38 number
roulette wheel will show all 38 numbers after 152 spins.

STEP 6
So, if we take the number
1.36 × 10240 (all results of spinning a 38 number roulette wheel 152 times)
and divide it by
6.72 × 10239 (all results of all 38 numbers appearing after 152 spins),
we get the probability of rolling all 38 numbers appearing after 152 spins.

Probability = 6.72 × 10239 ÷ 1.34 × 10240 = 0.501599170962349

Basically, you would have to spin a roulette wheel at least 152 times in order to have a better than 50 / 50 chance of spinning all 38 numbers.