A Math Formula That Actually Works

Now We’re Talking Gambling, Baby!

With gambling having preoccupied so many great minds, we would expect some of them to

have developed some mathematical formulas which would help in maximizing our

winning chances or, at least, minimize the hole left in our wallets when the game ends. And

we’d be right in assuming this.

One of the most popular mathematical formulas which can be used in this regard is the Kelly

Criterion. Going through its history a bit, this formula has been first published in 1956 and was

meant for any gambling context in which we have a positive expectation. If we were to put it

in simple language, we’d say that this formula can tell you how much you can bet without

risking going broke.

Basically, the Kelly Criterion implies using the highest possible bets without going broke. Now

let’s translate this into a proper formula:

 f* = (bp-q)/b,


  • f* = the fraction of the current bankroll to bet;
  • b = the odds received on the bet;
  • p = winning chances;
  • q = losing chances (1 – p).

If you look at it from a businessman’s point of view, you can calculate the return of investment

(ROI) as the wins minus losses, al divided by amount invested. Rephrasing the above formula in

such terms, we would now have:

f* = ROI * p / (ROI + 1 – p).

The Kelly betting strategy is meant to maximize the expected bankroll growth, in average luck

conditions. If you choose to bet a greater amount than what this formula indicates, then you can

get fatter gains, but you should know that a higher degree of volatility results in a lower long-

term growth.

A much more frequent strategy is choosing to bet less than the full amount indicated by the

Kelly formula. Although this tactic will lower your expected growth, it will also reduce volatility,

thus proving a more viable solution for many. For example, using a “Half Kelly” will double the

bankroll requirement, reducing volatility by 50%, but the growth by just 25%. This is

particularly recommended for people who do not feel comfortable with the bankroll used for

each buy-in level – this betting method poses less risks, so you can “play safe” with this one.

Now, there are multiple situations which may require you to apply this formula. For the simple

bets, with only 2 possible outcomes, the best Kelly bet can be found out by dividing the

advantage by what the bet pays in a to-one context. When we have more complex bets, a good

strategy is to use advantage/variance, where variance is calculated as the square of standard

deviation. The standard deviation is an indicator of the bankroll volatility. For a number of n

bets, it can be calculated as the product of the standard deviation for one bet (0.6826) and the

square root of the sum of initial bets.

All these details can be a lot to take in, especially for someone who is more into letters than into

numbers. If this is your case, then we recommend you to do some research on statistics – it will

help you a lot in understanding the terms and key concepts.

We hope you find this article helpful and wish you the best of luck!