

A355659


Martingale win/loss triangle, read by rows: T(n,k) = total number of dollars won (or lost) using the martingale method on all possible n trials that have exactly k losses and nk wins, for 0 <= k <= n.


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0, 1, 1, 2, 1, 3, 3, 5, 1, 7, 4, 11, 7, 7, 15, 5, 19, 24, 4, 21, 31, 6, 29, 53, 38, 12, 51, 63, 7, 41, 97, 111, 41, 57, 113, 127, 8, 55, 159, 243, 187, 5, 163, 239, 255, 9, 71, 242, 458, 500, 248, 130, 394, 493, 511, 10, 89, 349, 784, 1084, 874, 202, 488, 878, 1003, 1023
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OFFSET

0,4


COMMENTS

The martingale betting method is as follows: bet $1. If win, bet $1 on next trial. If lose, double your bet on next trial. Repeat for a total of n times.
We can use row n of the triangle to find the total expected value for n trials, if we assume that the probability of each win is p. The expected value is Sum_{k=0..n} T(n,k)*p^k*(1p)^(nk). In a "fair" game where p = 1/2, this equals 0, as expected.


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FORMULA

T(n,k) = T(n1,k) + T(n1,k1) + binomial(n1,k) for 0<k<n.
Sum_{k=0..n} T(n,k) = 0 (the sum of each row equals 0).
The following six formulas describe the three leftmost columns and the three rightmost diagonals of the triangle drawn below.
T(n,0) = n (this is the scenario with n trials, 0 losses; since the martingale method has us bet 1 after each win, we end up with total earnings equal to n).
T(n,1) = n^2  n  1 (this scenario is when there are n trials with just 1 loss; calculations show that this equals n^2  n  1 = A165900(n)).
T(n,2) = (n^3  3n^2  2)/2.
T(n,n1)= 1 + 2*n  2^n = A070313(n).
T(n,n2) = (3*n^2  n)/2 + 1  2^n.
G.f.: x*(1y)*(1x*y) / ((1  x*(1+y))^2 * (12*x*y)).  Kevin Ryde, Aug 30 2022


EXAMPLE

Triangle T(n,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9
+
0 0
1 1 1
2 2 1 3
3 3 5 1 7
4 4 11 7 7 15
5 5 19 24 4 21 31
6 6 29 53 38 12 51 63
7 7 41 97 111 41 57 113 127
8 8 55 159 243 187 5 163 239 255
9 9 71 242 458 500 248 130 394 493 511
Examples from triangle:
T(4,3) = 7: In this example, we consider all possibilities with 4 trials that result in 3 losses and one win. There are binomial(4,3) = 4 different combinations to consider (lllw, llwl, lwll, and wlll), which have net earnings of +1, 0, 2, 6 respectively when using the martingale method, giving a total of 7.
T(6,2) = 53: In this example, we have 6 trials and we consider the results with 2 losses and 4 wins. There are binomial(6,2) = 15 such combinations to consider (wwwwll, wwwlwl, wwwllw, wwlwwl, wwlwlw, wwllww, wlwwwl, wlwwlw, wlwlww, wllwww, lwwwwl, lwwwlw, lwwlww, lwlwww, llwwww), and summing over all 15 earnings gives us a total of 53.
T(2,0) = 2: In this example, we have 2 trials, with 0 losses and 2 wins. In this one single case, the martingale method gives us earnings of +1 and +1 with a total of 2.


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STATUS

approved



