Understanding the underlying math and its key components, such as probability, statistics, and expected value, is key to successful betting. That’s where betting strategies come into play, starting with the Kelly betting formula, aka the “scientific betting formula.”
Kelly Criterion gives you a roadmap to maximizing profits based on your bankroll and the value of your wagers in terms of edge and probability. This guide gives you a detailed explanation of the Kelly Criterion formula, examples of it, and instructions on how to use and modify it to make it work for your betting style.
Kelly Betting, or Kelly Criterion, is a mathematical formula developed by John L. Kelly Jr., a researcher at Bell Labs, that is used to determine the optimal size of bets to maximize long-term profits. In math terms, the formula stems from the concept of the expected logarithm of the return and should lead to the expected geometric rate of growth.
Essentially, this approach prioritizes the long-term growth of your bankroll over short-term gains, making it a robust strategy for consistent returns like no other. Unlike most other formulas, which were either developed for the stock market or other goals, the Kelly Criterion was first used for betting on horse racing in the 50s and then throughout the entire sports betting industry.
The core of this betting term is the Kelly Criterion formula:
𝑓∗=(𝑏𝑝−𝑞)/𝑏
Where:
With this formula, you can determine the proportion of your bankroll to wager on a particular bet to maximize your long-term growth of the bankroll.
Calculating the implied probability of the bet is an integral part of the Kelly criterion betting strategy since it defines the value of 𝑝. Getting this value requires careful analysis and often involves statistical models or historical data. Once you get the probability, you can go ahead and identify the odds.
The only issue here is that you need to convert these odds into decimal form and subtract 1 to get the value of 𝑏, which can be tricky if you are playing in US-oriented offshore sportsbooks where the American odds reign supreme.
For example, if the American odds are +150 when you convert them to decimal odds, you get 2.5 for the formula. When you subtract 1 from these, the value of 𝑏 would be 1.5.
Next up is calculate the Kelly fraction (f). Plug in the values of 𝑝 and 𝑏, and you’ll get the optimal fraction of the bankroll you need to stake. Finally, all you have to do is place a wager using the calculated fraction of your bankroll.
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As we have already seen, the Kelly betting system draws its strength from a mathematical formula. It all starts with the expected value of the logarithm of wealth. In that logarithm, the initial bankroll is 𝑊, and you bet a fraction 𝑓 of your bankroll. After the bet, your bankroll can either increase to 𝑊(1+𝑏𝑓) if you win or decrease to 𝑊(1−𝑓) should you lose.
The expected logarithm of wealth is then: 𝐸[log(𝑊𝑛𝑒𝑤)]=𝑝log(𝑊(1+𝑏𝑓))+𝑞log(𝑊(1−𝑓))
The components of the equation are as follows:
Now, we need to find the value of 𝑓 that maximizes 𝐸[log(𝑊𝑛𝑒𝑤) using the derivative of the expected logarithm of wealth/bankroll concerning 𝑓. By setting the derivative to zero for optimal results, we get the final formula:
𝑓∗=(𝑏𝑝−𝑞)/𝑏
Let’s introduce a practical example to illustrate the Kelly Criterion in action. Suppose you have a bankroll of $1,000, and considering a bet on the Chiefs taking down the 49ers with decimal odds of 3.00 (+200). These odds imply a potential profit of $2 for every $1 bet. If you estimate the probability of winning this bet at 50%, despite the implied probability stating 33%, that would be a good chance to use the formula. The calculation for the Kelly Criterion in football bets like these would be as follows:
𝑓∗=((2×0.50)−0.50))/2=(1−0.50)/2=0.50/2=0.25
According to the Kelly staking plan, you should wager 25% of your bankroll, or $250, on this bet. If you decide to bet more than 25% of your bankroll, you’ll be taking on way too much risk. If you go under that threshold, you won’t be taking full advantage of the edge.
The Kelly bet sizing strategy is best seen through actual examples. Say we want to make a parlay bet on two NFL games: the Green Bay Packers and the Kansas City Chiefs both winning against the Los Angeles Rams and San Francisco 49ers, respectively. The bookmaker offers combined decimal odds of 4.00 (+300) for the parlay. You estimate the probability of both events happening at 30% rather than the 25% implied probability.
Using the Kelly calculator with the following information:
f ∗ = (3.00×0.30−0.70)/3.00 = (0.90−0.70)/3.00 = 3.00/0.20 =0.0667
We should wager 6.67% of your bankroll or $66.70 of our $1,000 balance on that parlay bet.
On the other hand, the Kelly criterion sports betting formula can also tell us if the bet is not worthy enough.
For example, you want to bet on an NBA game between the Los Angeles Lakers and the Golden State Warriors. The bookmaker offers the Lakers at -110, meaning the probability there is 52.5%. However, you think that LeBron James and Bronny James, his son, will be a killer duo in that one, and you estimate the probability of the Lakers winning is at 55%.
Using the Kelly formula, you get the following values:
f∗= 0.80×0.55−0.45/0.80=(0.44−0.45)/0.80=−0.01/0.80=−0.0125
A negative value indicates that the bet is unfavorable according to the Kelly ratio and should be avoided or we should go with a little less aggressive alternative — the Partial Kelly or the Quarter Kelly.
The Partial Kelly is a common variation of the Kelly Criterion where the aim is to reduce variance by lowering the bet size, usually dividing it by 2.
It is used primarily because the full Kelly can be difficult to sustain in regards to the bankroll. In a student journal of The Wharton School of the University of Pennsylvania, Philadelphia dubbed An Investigation of Sports Betting Selection and Sizing, the theory was tested, and even though it proved unsuccessful, meaning the bankroll was exhausted in the end, still, the ruin of the bankroll came at a later stage than when going with the full Kelly.
There is an even more cautious Kelly criterion example, known as the Quarter Kelly. The Quarter Kelly lowers the risk even further since you’ll be betting just a quarter of what the full Kelly has suggested. In fact, the team of the Wharton School of the University of Pennsylvania recorded a profit of $40,374 over 10,275 bets, and the profits grew faster as they lowered the betting coefficients.
This is not the first time a similar test took place. In 1994, Edward O. Thorp, an American mathematician and professor, tested the Kelly Criterion and recorded his findings in a scientific paper, which is also one of the reasons why this is regarded as a scientific approach to sports betting.
The Kelly Criterion formula has its clear advantages but doesn’t come without risks. Let’s take a look at its pros and cons.
The Kelly strategy is a completely legal betting strategy. Although online betting sites, especially new bookmakers, may frown on its usage since it could bring players additional security and more prudent betting patterns, there is nothing about the formula or the Kelly Calculator that makes it illegal.
In fact, the operators don’t even have to know you are using the Kelly strategy. How you manage your bankroll is your business and your business solely. However, note that should the bookies notice that you’ve been constantly increasing your bankroll, they may flag your account and reduce your betting limits. Bookies are known to limit those who are on a roll, which is why you should be careful about using the method too much.
The scientific and systematic approach to the method is what makes the Kelly Criterion a worthy strategy. It helps you determine the optimal portion of your bankroll to wager on a given bet, based on the probability of winning and the odds offered. This way, you ensure that you have enough bankroll to keep betting and hopefully turn the tide if things go south.
The accuracy of your probability estimates is crucial in leveraging the formula and strategy. You need to use predictive models and algorithms to enhance the accuracy of your probability assessments and find those value bets. Look for instances where you believe the bookmaker’s odds undervalue a team’s chances. This is where the Kelly Criterion can be particularly effective.
On the other hand, Kelly betting can be tricky for sports betting as it is such a dynamic environment, especially live betting. You can adapt it for in-play betting, but you’ll have to continuously update probability estimates based on the line movements, and that can be too difficult to trace.
The formula can also be used on casino games, and with potentially greater success. Namely, new online casinos offer games with fixed odds and probabilities, so there won’t be many adjustments. Also, with casino gaming, the emotional element and the media hype are not as potent as it is when it comes to sports betting.
No betting strategy is foolproof, including Kelly Criterion. However, here are some tips that sports bettors can use to get the most out of it.
Kelly betting is not the only sports betting strategy out there. Bettors also frequently opt for the Martingale system where they double the bet size after each loss, or the Fibonacci betting system based on the Fibonacci sequence.
The Labouchere system is a negative progression betting system where you increase the stake with every consecutive loss until you recover the balance with a win. There’s also the Paroli betting system, frequently labeled as safer than the Martingale system.
The most important point is to find the betting method that best suits your bankroll and betting style.
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To gamble responsibly means to be in complete control of your spending habits, not chasing losses, and no matter how good a betting system may sound, don’t follow it blindly. We encourage you to use spending and deposit limits and other tools to stay in control of your finances. That way, you can have the best possible betting experience.
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