A losing run is defined as a number of consecutive losing bets. I've written a program to simulate this but the results I'm getting are different from what the formula gives, so one or another is wrong (I suspect the formula).
Here is how I derived the formula :
The chance of winning the first bet betting x$x$ numbers is x/37$\frac{x}{37}$
So the chance of losing it is (37 - x)/37$\frac{37 - x}{37}$
By the Geometric distribution, the chance of a losing run of length r is
(1) [(37 - x)/37]^r * (x/37) $(\frac{37 - x}{37})^r * (\frac{x}{37})$
Now since I'm only concerned with the losing bets, the number of losing bets in the sample n$n$ is
(37 - x)/37 * n$\frac{37 - x}{37} * n$
And these losing bets consist of the sum of all the losing runs of length 1,2,3,...$1,2,3,\ldots$
So to find the number of losing runs of length r, I need to multiply the number of losing bets by (1), which is
[(37 - x)/37]^(r+1) * (x/37) * n$(\frac{37 - x}{37})^{r+1} * \frac{x}{37} * n$
So for example, if x = 12$x = 12$ (betting a dozen section on the layout), n = 10,000$n = 10,000$, and r = 5$r = 5$, I get
(25/37)^6 * (12/37) * 10,000 ~ 309$(\frac{25}{37})^6 * \frac{12}{37} * 10000 \approx 309$ losing runs of length 5.
However, my simulation output is:
Sum of Loss Streaks = 6751
1 744
2 456
3 339
4 188
5 153
6 98
7 77
8 34
9 35
10 24
11 12
12 11
13 6
14 2
15 2
16 2
17 3
18 2
19 3
20 0
21 0
22 0
23 0
24 0
25 0
26 0
27 0
28 0
29 0
30 0
31 1
There are 153 loss streaks of length 5, so I seem to be out by a factor of about 2.
Thanks in advance for any help.
