Gambling math question for Ganch asked by Peep

[raiders72002]

Peep

One number I would like to know and can't find is

1) If you know you win _____________________%
2) At odds of ______________________.
2) And you bet _________________________% of your bankroll.
3) What percentage of the time will you go broke? What is your percentage chance of ruin?

[durito]

There's not enough info to answer the question. Like, over how many bets.

But, if you are betting a % of your bankroll (adjusting after every bet and never have 100% risked at one time) you can't go broke. You may get a sufficiently small enough bankroll that you fall below betting limits.


Ganch made a spreadsheet though that does this calculation. I'm sure it can be found through a search.

[Ganchrow]

One number I would like to know and can't find is

1) If you know you win _____________________%
2) At odds of ______________________.
2) And you bet _________________________% of your bankroll.
3) What percentage of the time will you go broke? What is your percentage chance of ruin?

It is not a hard formula, I did have it in an old horse racing book, but I lost it.

For example, say
1) You can win 70% of your bets at -140. What percentage of your bankroll can you bet to give yourself only a 2% chance of going broke.

I'm actually not quite sure I understand your question here, Peep. If you're betting a percentage of your bankroll < 100% then your chances of "going broke" in the sense of losing everything will always be exactly zero.

Think about it this way ... no matter how much you slice up a pie then as long you leave some of it on the table then after a finite number of slices you'll always have at least a little sliver of pie remaining.

So that said, it doesn't strictly make sense to talk about the probability of "going broke" in the context of Kelly or indeed of any percentage of bankroll betting scheme. A more meaningful question might be one to the effect of:

You can win 70% of your bets at -140. What percentage of your bankroll can you bet to give yourself only a 2% chance of your bankroll dropping to 1% or less of its initial value over 500 trials.

Such a question can easily be answered exactly using the binomial distribution. Assuming a starting bankroll of 1:

Let N = # of bets
Let W = # of winning bets
Let X = fraction of bankroll wagered per bet
Let d= decimal payout odds
Let log(B) = the natural logarithim of bankroll after N trials
Let p = single-bet probability
let α = selected "ruin" probability

So:

log(B) = W*log(1+(d-1)*X) + (N-W)*log(1-X)

To determine the critical number of wins from the binomial distribution such that the probability of "ruin" is ≤ α we can use the Excel CRITBINOM() function:

W* = CRITBINOM(N, p, α)

So plugging in the figures from above we have:

W* = CRITBINOM(500, 70%, 2%) = 329

and

log(1%) = 329*log(1+0.714285714*X) + (500-329)*log(1-X)

Solving for X gives us 39.9377% (which is quite a bit higher than the full-Kelly stake of 28%).

This tells we'd need to risk 39.9377% of bankroll per wager at -140 given a 70% win probability to have a 2% probability (it actually works out to about 1.89%) of our bankroll dropping to 1% or less of initial over 500 bets.

One problem with this above solution, however, is that it only gives the probability of finishing at some % of initial bankroll rather than of attaining some results at any point over N trials. By appealing to the Brownian motion equations, Thorp (of Beat the Dealer fame) gives a derivation of the above that may be of use to the sports bettor. You can read his paper here.

[raiders72002]

Peep

Yes, that would be a good question gauchrow, and basically the one I want to ask. If and when I start gambling again, I will play with it.

I think your answer works.

And I think that is the same question I am trying to frame. I want to know what is my optimum bet to maximize profits, but do not want to risk more than a certain percentage of chance of tapping out (which being down to 1% effectively is). 2% chance is probably one most gamblers, including myself, could live with.

[HedgeHog]

In a round about way, this is basically a Kelly question. Assuming Peep is a +Ev better, he has two serious risks. He could bet too much per game, wiping out his advantage with one bad streak. Or he colud bet too little on games he has a much larger advantage on, thus leaving money on the table.

[ManOfValue]

Ganchrow, is it ok if I call you "MY HERO" ???

[Ganchrow]

Ganchrow, is it ok if I call you "MY HERO" ???

Ok by me, but you'll still have to get prior written approval from my wife on that one.