The probability that any given number will not have hit is (37/38)200 = 0.48%.
With 38 numbers, we could incorrectly say that the probability that any one of them would not be hit is 38 × (37/38)200 = 18.34%.
The reason this is incorrect is it double counts two numbers not being hit. So we need to subtract those probabilities out. There are combin(38,2) = 703 sets of 2 numbers out of 38. The probability of not hitting any two given numbers is (36/38)200 = 0.000020127. We need to subtract the probability of avoiding both numbers. So we are at:
38×(37/38) 200 - combin(38,2)×(36/38) 200 = 16.9255%.
However, now we have canceled out the probability of three numbers not hitting. For any given group of three numbers we triple counted the probability of any single number not being hit. We then triple subtracted for each way to choose two numbers out of the three, leaving with zero for the probability that all three numbers were not hit. There are combin(38,3)=8,436 such groups. Adding them back in we are now at:
38×(37/38) 200 - combin(38,2)×(36/38) 200 + combin(38,3)×(35/38)200 = 16.9862%.
Yet, now we have over-counted the probability of four numbers not hitting. For each of the combin(38,4)=73,815 groups of four numbers, each was originally quadruple counted. Then we subtracted each of the combin(4,2)=6 groups of 2 out of the 4. Then we added back in the 4 groups of 3 out of the 4. So, for each union of four numbers, it was counted 4 – 6 + 4 = 2 times. To adjust for the double counting we must subtract for each group. Subtracting them out we are now at:
38×(37/38) 200 - combin(38,2)×(36/38) 200 + combin(38,3)×(35/38)200 - combin(38,4)×(34/38)200 = 16.9845%.
Continuing in the process we would keep alternating adding and subtracting, all the way until missing 37 numbers. Thus the probability of at least one number never being hit is:
Sum i=1 to 37 [(-1)(i+1) × combin(38,i) × ((38-i)/38)38] = 16.9845715651245%
Here are the results of a random simulation of 126,900,000 such 200-spin experiments.
| Numbers Hit in 200 Roulette Spins | ||
| Numbers Hit | Observations | Ratio |
| 31 or Less | 0 | 0 |
| 32 | 1 | 0.00000001 |
| 33 | 33 | 0.00000026 |
| 34 | 1812 | 0.00001428 |
| 35 | 68845 | 0.00054251 |
| 36 | 1577029 | 0.01242734 |
| 37 | 19904109 | 0.15684877 |
| 38 | 105348171 | 0.83016683 |
| Total | 126900000 | 1 |
The ratio of times at least one number was not hit was 0.169833.
Thanks to J.F.W. from Marshall for this problem.
Michael Shackleford, A.S.A.