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:

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):

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.