Let's suppose you bet on an event with 75% chances to win. If you lose you lose 1 bet, if you win you win 1 bet. So the risk:reward ratio is 1:1
You have a starting amount of money and are allowed to bet whatever amount do you want but if you lose all your money you're out of the game.
What % of that amount would you bet? 50% would be too much, 1% would be too less.
You are allowed to make that bet only 3 times a month, so you can't bet whenever you want (risking a tiny % because you can make that bet non stop won't work, you can make it only 3 times a month).

This is a basic kelly criterion question. The kelly criterion tells you what fraction of your bankroll to bet to maximize bankroll growth. This is somewhat aggressive for some people so they will sometimes use a fraction of the result of the KC, i.e. "fractional kelly". Your situation is the easiest kind of calculate.

The KC is
f = (bp - q) / b
where b is the payout in b:1 format, p is the chance to win, q is the chance to lose (1-p usually)
Here the payout is 1:1 so b = 1, therefore
f = p - q = p - (1-p) = p - 1 + p = 2p - 1
Since p = 75%, f = 50%

50%? But the risk of ruin is high. I want the RoR to be 0 or close to 0. If I lose all my money I'm out of the game.

The risk of ruin at 50% of bankroll is very low. Almost 0 - well under 1% (Edit: the actual risk of ruin is 0 if you're allowed to keep betting arbitrarily small amounts of money. If there's a minimum bet, then the ROR is dependent on the ratio between that and your bankroll)

Say you start with 1000. Your first bet is 500. If you lose, your next bet is 250. Do you see how it's very hard to go broke?

I suck at math but I wrote a quick simulation. Betting 50% of your bankroll, let's assume that the minimum bet size is 1 unit, i.e. you are broke if you ever have less than 1 unit. Here is the RoR for various BR sizes:

1000: .04%
100: .67%
10: 7.5%

So you can see even with a very modest bankroll compared to the minbet size, your RoR is not bad.

I thought my next bet would be 500, not 250. Now I understand.

Now I will add another rule:

I am only allowed to bet fixed amounts. That fixed amount is a % of my initial bankroll.

How big should be that % in order to RoR to be 0 or close to 0 if the reward:risk is 1:1 and the probability is 75%

How about if the reward:risk ratio is only 0.5:1?

It all comes down to the kelly criteria. This is used often and extensively, like hedge funds.

Okay now it depends on two things:
1) Are you going to quit after a predefined profit?
2) Are you asking the risk of EVER going broke, or going broke in some finite number of bets N?

No. I'm not going to quit.
I want to risk fixed amounts (for instance 10% of my initial bankroll) if my bankroll is smaller or equal to my initial bankroll and to risk a % of my current bankroll only if my current bankroll is bigger than my initial bankroll.
For instance i start with $100 000. I will start to risk let's say $10 000 and if my bankroll is smaller than $100 000 I will continue to risk $10 000 until I will go broke.
However if I will have $110 000 I will risk 10% of 110 000.

The win rate is 75% and the payoff is either 0.5:1 or 1:1.
I want to find what's a good % of my initial bankroll for the case in which my current bankroll is smaller or equal to my current bankroll.
I want to use fixed amounts because if I will be unlucky it will be hard to recover if I continue to use a % of my current bankroll.

I'm asking these questions for forex trading.

I deleted your other thread. Keep the discussion here. You're asking many different questions which all have very different answers. First you're asking about betting a percentage of your bankroll. Then you're asking about betting a fixed amount. Then you're asking about betting a variable percentage only if you win. Then you change from 1:1 payoffs to maybe 0.5:1 payoffs. These are huge differences. They all require time and effort to analyze properly, and they must be analyzed differently. It's hard to do that when the question isn't clear and changes every 5 minutes.

There's a world of difference between having to bet a fixed amount and being able to bet any amount. Betting a fixed amount has nothing to do with the Kelly criterion. If you played forever getting 1:1 payoffs as a 3:1 favorite, your risk of ruin is simply (1/3)^n where n if the number of bets in your bankroll. So if you bet half your bankroll on the first bet, and always bet that same dollar amount, your risk of ruin would be 1/9. If you bet your whole bankroll on the first bet and stuck with that dollar amount, it would be 1/3. But your actual risk of ruin will be less than this if you quit after some period of time. This all assumes that you never change your bet. If you increase your bets only when you are ahead, your risk of ruin will be higher than this.

If you could bet any amount you wanted on each bet getting 1:1 payoffs, then the Kelly criterion tells you that betting 50% of your bankroll every time will maximize your exponential rate of bankroll growth. But you have to be careful if you only play a limited number of times because there is a significant chance that you will lose money, and when you lose money, your minimum loss can be about half your bankroll. If you played 12 times, you would have about a 16% chance of losing over 47% of your bankroll. You will have close to a 50% chance being down half your bankroll at some point if you played forever. If you played half Kelly betting 25% of your bankroll, your chance of losing money after 12 plays would be over 5%, but when you lose, almost 75% of the time you will come away with 2/3 of your bankroll. On the other hand, if you played 150 times, the chance of losing money betting full Kelly would be miniscule at 0.055%, and the factor by which your bankroll would grow if you won 75% of your plays would be about a trillion. Compare that to the average 1 bet per play you get by fixed betting. This just comes from the binomial distribution and computing your final bankroll as

B*(1 + f)^W * (1 - f)^L

B = starting bankroll
f = fraction of bankroll to bet
W = wins
L = losses

If you are only getting 0.5:1, then your Kelly fraction would be 25%, and your final bankroll is computed from

B*(1 + 0.5*f)^W * (1 - f)^L


Compute your Kelly fraction from

f = p - q/b

f = your Kelly fraction
p = your probability of winning
q = 1-p
b = your payoff odds (1 for 1:1, 0.5 for 0.5:1)


When you bet Kelly, you maximize your exponential rate of bankroll growth at the expense of having a higher probability of at some point reducing your bankroll to any given percentage of its starting value compared to fixed betting. If you play forever, your probability of reducing your bankroll to 50% of its starting value at some point is 50%, to 25% is 25%, to 10% is 10%, and so on. You have a 33% chance of halving your bankroll before doubling it.

What is the formula for 2:1 or 0.5:1 payoffs?
I think you used ((1-edge) / (1+edge))^n

For 1:1 the edge is 0.5
But for other payoffs how do I calculate edge?

I like u man, your smart