Ever since I started working on DiceBot I've been thinking about how to calculate the idea multiplier for a certain chance to win, because I noticed that for everything except x2 payout, the obvious settings does not work that well.

For example, when betting at 25% chance to win, the first thought is that you should increase your bet by 25% after each loss, or to use an equivalent multiplier of 1.25, to maintain a constant profit after your eventual win. Except that it doesn't work. Take a look at the table below:

No	Bet	Total Wagered	Bet Profit	Actual Profit
1	1	1	3.96	2.96
2	1.25	2.25	4.95	2.7
3	1.5625	3.8125	6.1875	2.375
4	1.953125	5.765625	7.734375	1.96875
5	2.44140625	8.20703125	9.66796875	1.4609375
6	3.051757813	11.25878906	12.08496094	0.826171875
7	3.814697266	15.07348633	15.10620117	0.032714844
8	4.768371582	19.84185791	18.88275146	-0.959106445
9	5.960464478	25.80232239	23.60343933	-2.198883057
10	7.450580597	33.25290298	29.50429916	-3.748603821

As you can see, after a few losses in a row (8), you start losing instead of making profit, and at 25% chance to win, you will regularly have more than 8 losses in a row. I've always just kind of winged it and set it to much higher than it needs to be, but this is riskier because the bets become much larger much quicker in a long losing streak. And I've never bothered doing the math to figure out what it should be.

The Math

So today I sat down and did it. And here's what I came up with:1.3378378378378378378378378378378
That is the ideal multiplier for 25%.

The formula to calculate the idea multiplier for any chance to win or payout is extremely simple and I cannot believe it took me this long to sit and and figure out it. The formula is simply:

Multiplier = Payout / (Payout -1)

From one of my previous articles DIceBot Math I explain how to convert between chance to win and payout. So using that formula, converting from 25% chance to win, on a site using 1% house edge:

payout = (100-1)/25=3.96

So the ideal multiplier would be :

Multiplier = 3.96/(3.96-1)=3.96/2.96=1.3378378378378378378378378378378

Converting this back to chance increase gives you an increase of 33.78378378378% increase on each loss, not 25% as one would expect.

And now, the profit stays constant much much longer (it will start to increase/decrease a bit eventually because of precision limitations imposed by computers and operating systems):

No	Bet	Total Wagered	Bet Profit	Actual Profit
1	1	1	3.96	2.96
2	1.337837838	2.337837838	5.297837838	2.96
3	1.78981008	4.127647918	7.087647918	2.96
4	2.394475648	6.522123566	9.482123566	2.96
5	3.203420124	9.72554369	12.68554369	2.96
6	4.285656652	14.01120034	16.97120034	2.96
7	5.733513629	19.74471397	22.70471397	2.96
8	7.670511477	27.41522545	30.37522545	2.96
9	10.26190049	37.67712594	40.63712594	2.96
10	13.72875876	51.4058847	54.3658847	2.96
11	18.36685294	69.77273764	72.73273764	2.96
12	24.57187082	94.34460846	97.30460846	2.96
13	32.87317853	127.217787	130.177787	2.96
14	43.97898209	171.1967691	174.1567691	2.96
15	58.83674631	230.0335154	232.9935154	2.96
16	78.71402547	308.7475409	311.7075409	2.96
17	105.3066016	414.0541425	417.0141425	2.96
18	140.8831563	554.9372988	557.8972988	2.96
19	188.4788172	743.4161159	746.3761159	2.96
20	252.1540932	995.5702092	998.5302092	2.96
21	337.3412869	1332.911496	1335.871496	2.96
22	451.3079379	1784.219434	1787.179434	2.96
23	603.7768358	2387.99627	2390.95627	2.96
24	807.7554965	3195.751766	3198.711766	2.96
25	1080.645867	4276.397633	4279.357633	2.96
26	1445.72893	5722.126563	5725.086563	2.96
27	1934.150866	7656.277429	7659.237429	2.96
28	2587.580213	10243.85764	10246.81764	2.96
29	3461.762717	13705.62036	13708.58036	2.96
30	4631.277148	18336.89751	18339.85751	2.96
31	6195.897806	24532.79531	24535.75531	2.96
32	8289.106525	32821.90184	32824.86184	2.96
33	11089.48035	43911.38219	43914.34219	2.96
34	14835.92642	58747.3086	58750.2686	2.96
35	19848.06372	78595.37232	78598.33232	2.96
36	26553.49065	105148.863	105151.823	2.96
37	35524.26452	140673.1275	140676.0875	2.96
38	47525.70523	188198.8327	188201.7927	2.96
39	63581.68673	251780.5194	251783.4794	2.96
40	85061.9863	336842.5057	336845.4657	2.96
41	113799.1438	450641.6496	450644.6096	2.96
42	152244.8005	602886.4501	602889.4101	2.96
43	203678.8548	806565.3049	806568.2649	2.96
44	272489.2787	1079054.584	1079057.544	2.96
45	364546.4674	1443601.051	1443604.011	2.96
46	487704.0578	1931305.109	1931308.069	2.96
47	652468.9421	2583774.051	2583777.011	2.96
48	872897.6388	3456671.69	3456674.65	2.960000001
49	1167795.49	4624467.179	4624470.139	2.960000001
50	1562320.993	6186788.173	6186791.133	2.960000001

This works for all payouts. Lets take the most iconic one, 49.5% chance to win, or x2 payout:
Multiplier = 2/(2-1) = 2/1= 2
Or, a 100% increase on loss.

For 95% chance to win:
Payout = (100-1)/95=1.1042105
Multiplier=1.042105/0.042105=24.75

For those interested, here's a LUA function you can use in the programmer mode to automatically calculate the multiplier:

function getmulti()
payout=(100-1)/chance
return payout/(payout-1)
end

--You can use this like follows:
nextbet=previousbet*getmulti()

I hope you find this information useful in your gambling endeavors.