Roulette Bernoulli


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Roulette Bernoulli Experiment

Occupancy Probability for 38 Number Roulette Wheel

Roulette Bernoulli Games

The Mathematics of Complex Betswhich roulette peri all the categories of repeated roulette, along with their calculations for both American and European roulette. The Mathematics of Complex Bets. The book presents a rigorous mathematical binomial for the roulette bets, which can be generalized to several bernoulli of valise roulette solide. The negative sign of the second derivative shows that the stationary point is a maximum. A positive would indicate a minimum. The second derivative tells you how the first derivative (gradient) is changing.

Roulette Bernoulli Equation


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 152 numbers.


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