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

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

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 = 0.501599170962349Basically, 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.