Need math help to solve for how to rip off gambling sites

[ChpBstrd]

I am not a gambler, but I've noticed a lot of sports betting sites and horse racetracks now offer betting insurance - sometimes up to $1000. That means if you lose your first bet, they will allow you to make another bet for free and keep the proceeds if that second bet wins.

Their goal is obviously to addict new customers, but I wonder if there is an opportunity to play this game strategically and systematically across many sites and obtain a positive expected value by carefully selecting odds. This is similar to how credit cards are generally an awful deal, but can be played to obtain four-figures worth of bonuses per year while paying no interest. There are enough gambling sites offering first bet insurance that a person could run through dozens of them and probably make a decent net profit across them all IF they could find the mathematically optimal betting strategy.

With sports, casino games, and horses, one can generally choose the odds at which to bet. E.g. bet on an unlikely winner for either a big potential payout or small loss, or bet on a likely winner for either a small potential payout or big loss.

So I wonder, for your first bet is it optimal to bet on a team/horse/hand that has a high (e.g. 80%) chance of winning a small amount, cash out if you win, and if you lose the first bet then bet on another likely outcome for your second bet? Or is it more optimal to swing for the fences on that first bet, and if you lose make your second bet a high-probability one to probably recover your money?

At a superficial level, there are two ways to win and one way to lose:

1) Win first bet and cash out.
2) Lose first bet, win second bet, cash out.
3) Lose both first and second bets.

Yet one can vary the odds of both the first and second bets, which adds complexity.

This is where my memory of probabilistic math fails me.

If all bets were coin flips with 50/50 odds, double or nothing, then the possible outcomes would look like:

1) Probability of winning first bet and cashing out = 50%
2) Probability of losing first bet = 50% and the conditional probability of winning second bet = 50% as a combination equals (0.5 * 0.5 =) 25%.
3) Probability of losing both coin toss bets equals (0.5 * 0.5 =) 25%

So in this example, the odds of doubling your money are scenario 1 + scenario 2 = 75% and the odds of losing all your money is scenario 3 or 25%. Based on this little exercise alone, it looks like there is profit to be made from these offers.

So we want to solve for the odds of winning at bets of various probabilities by simply continuing this process down our spreadsheet for bets with odds different than our 50/50 example.

However this method will simply tell us to take all the safest bets, with no accounting for the fact that the payouts are higher for risky bets than for safe bets, and not accounting for the house's cut. It is possible to win very frequently and still not make money if the payouts are low enough and you lose your entire bet some percentage of the time. E.g. if we make our 2 bets have an 80% chance of winning, the odds of winning one of them are 96%.

pW1 = probability of winning first bet
pL1 = probability of losing first bet
pW2 = probability of winning second bet

(pW1 + (pL1*pW2)) = (.8 + ((1-.8)*.8)) = 96%.

However if the house's cut on the payout is 5% for example, then we might have lost money by winning this game! We can envision the house's cut as a percentage taken off the payout if a win occurs.

If for example we lower the odds of each bet to 40%, we obtain a 64% chance of winning at least one of them. Then we need to figure out what the payout would be for a bet with 40% odds (or any odds). There is a 64% chance we receive this payout, and a 36% chance we lose our bet. Because I am not a gambler, this is where my grasp on the math gets fuzzy and I need your help.

According to my read of this site, we should expect a true odds ratio of (p / (1-p)). So if there is a 40% chance of winning, the Odds Ratio would be (0.4/(1-0.4)=) 0.67 or 2 to 3. That is to say, if 2 people put in a hundred dollars each betting this way and 3 other people put in a hundred dollars each betting against them, the people betting $200 could win the total payout of $500 dollars (getting their money back plus the other people's money) which is a profit of $300. Vice versa for the other side of the bet if they win.

So for our spreadsheet's purposes, the Profit of a winning bet is the formula = 1/OddsRatio. Total Payout is Profit+Bet, or Profit + 100%. Profit at various win probabilities with no house cut included is:

80%: 0.25x
60%: 0.67x
50%: 1x
40%: 1.5x
20%: 4x

So I built a spreadsheet to find the combination of 2 conditional bets with the highest expected value, with a given house cut. My formula for the EV of a winning bet looks like:

(OddsFirstBet*PayoutFirstBet)+(OddsSecondBetGivenLossOnFirstBet * PayoutSecondBet) - HouseCut

OddsFirstBet * PayoutFirstBet reduces to one because it's a fair bet with no house profit accounted for yet.
OddsSecondBetGivenLossOnFirstBet is ((1-OddsFirstBet)*OddsSecondBet).

The EV of a double losing bet is simply -100%.

The total EV of the game is the expected value of a win times the probability of winning either bet plus the EV of losing times the probability of losing.

Conclusions:
Somebody please check my math because I have little confidence in my work.

Using a 10% house cut and 20% odds increments, initial results suggest that most combinations of odds have an EV less than 100% and are thus bad decisions. However, four combinations of two 20%, 40%, 60%, or 80% bets stand out as profitable decisions. For example, betting at 1:4 (20%) odds for your first bet and then 4:1 (80%) odds for your second bet yielded a total game EV of 126.8%. Thus if you repeated this play enough times you'd probably earn a 26.8% profit.

Of course the usual implementation caveats apply. What are the rules for withdrawing funds after gambling one or two rounds? Are the most profitable bet combinations even made available to betters doing the promotion? Are these games rigged? Will you get addicted and wreck your life instead of ripping off online casinos for a few hundred bucks? Will the bets have to be different sizes? Are the other players who set the odds rational experts? Is it worth the time? Etc.

For now I want to leave these concerns behind and focus on whether the mathematical system is right. Is it? Please check my attached spreadsheet and let me know your thoughts. If this is the correct approach, I'll expand it down to the percent to see what the optimal bet is.

[Sibley]

This is how you become a gambler.

[diffusate]

I made some really good money (50k or so) more than 20 years ago doing something very similar with the initial round of online casinos. They all had signup bonuses with minimum bet amounts that statistically ended up ahead. The details really do matter and are deep in policies and terms of service. I actually lost all interest in gambling after this because it's no fun when you no longer have the advantage on your side. Just keep good track and start small to see how it goes.

[GilesMM]

You can’t win if you don’t play!

[GuitarStv]

This is how you become a gambler.

But he's got a system!  You can't lose if you have a system!  :P

[merula]

But he's got a system!  You can't lose if you have a system!  :P

Certainly it depends on the system, and whether that system requires ping pong balls to remember whose turn it is.

I used to work with a guy who started his career in the staid financial sector I work in, then went to work for a Vegas casino for a few years to use his analytics skills, and then came back to the staid sector. He had great stories about how certain models of slot machines have visible indicators of when they are going to pay out imminently, so if you have that industry knowledge you can make plenty of money implementing it. But it pretty quickly ceases to be fun, to the point that a person with that knowledge opted to work in data analytics in a niche part of a staid financial sector.

[By the River]

I see all those ads as well.  My "plan" would be to bet the maximum covered bet on some 3 or 4 leg parlay during the first week of the college football season.  If I win, cash out.  If not, bet the money line that Georgia would beat Tennessee Tech the second week and then cash out (probably for a penny or two more than I put in).    I've never read the fine print to see if that's do-able and also never signed up for any sites.   
I'll leave the mathematical proof of this concept to others.

[reeshau]

The sports books have it covered.  They would see your system as an "obvious error."

https://www.washingtonpost.com/sports/2024/01/17/sports-bets-errors-payouts/

From the article, you might have a better chance if you are in NJ.

[plog]

(OddsFirstBet*PayoutFirstBet)+(OddsSecondBetGivenLossOnFirstBet * PayoutSecondBet) - HouseCut

Congratulations, you just independently invented professional sports gambling.  How do you know what the Odds are?  That's both a direct and rhetorical question.

Normally the sports book sets the odds of events which they do by setting the payouts for certain bets.  Your equation above has Odds and Payouts as unique variables, which means you have separated what the sports book thinks the odds are are and what the true odds are.   

The essence of professional sports gambling isn't knowledge or fandom of sports or horses, it's exploiting inefficiencies in the betting markets.  And after all the math and explanations that's what your post comes down to--how do you find the events on which there are incorrect odds.  That analysis hole is way deeper than what you have already done.

Let's talk math though--game theory.  Without knowledge of the true odds, we must accept the casinos odds as the true odds.  With that, no matter the odds or the bet, your expectation of any single bet is $0--that's the definition of true odds. I pay you $5 if you roll a 5, you pay me $1 if you roll anything else: 5*$1 = 1*$5.  But when you add in that if you lose on your first roll, you get to roll again for free, that shifts things into your favor. 

So that promotion is positive EV, but because that second bet has true odds as well, it doesn't lend itself to a maximization strategy other than betting the maximum you can. At which point you meet every gambler's friend--variance.

[ChpBstrd]

(OddsFirstBet*PayoutFirstBet)+(OddsSecondBetGivenLossOnFirstBet * PayoutSecondBet) - HouseCut

Congratulations, you just independently invented professional sports gambling.  How do you know what the Odds are?  That's both a direct and rhetorical question.

Normally the sports book sets the odds of events which they do by setting the payouts for certain bets.  Your equation above has Odds and Payouts as unique variables, which means you have separated what the sports book thinks the odds are are and what the true odds are.   

The odds are only knowable by the betting behavior of the herd. For things like horse races, the number of people betting on a horse sets the odds for that horse, offering a pot of money to be won by whomever is willing to bet against it.

If the betting company is setting the odds on both sides, they must set them fairly efficiently or else they'll be unable to attract bettors while simultaneously earning the house's cut. E.g. if the herd knows the probability of an event is 50%, but the company's odds suggest it is 60% and there is no way to bet against the company, then fewer people will make bets with that company versus a competitor offering better odds/payouts on the same bet. Similarly if a company's odds-setters are often in disagreement with the crowd, the company is likely to lose any of its own money it bets. If the company is still around, they are probably efficient. People absolutely watch multiple sites for a little gap in the odds so they can bet on the Lakers in one site and against the Lakers in the other site and earn a risk-free profit.

Of course, I noted an open question about whether people will be allowed bets with certain odds in their bonus round, or if the hand-selected bets made available to them exclude the bets with the best combined odds. This would be a second question to research after figuring out the mathematically optimal approach.

The essence of professional sports gambling isn't knowledge or fandom of sports or horses, it's exploiting inefficiencies in the betting markets.  And after all the math and explanations that's what your post comes down to--how do you find the events on which there are incorrect odds.  That analysis hole is way deeper than what you have already done.

Let's talk math though--game theory.  Without knowledge of the true odds, we must accept the casinos odds as the true odds.  With that, no matter the odds or the bet, your expectation of any single bet is $0--that's the definition of true odds. I pay you $5 if you roll a 5, you pay me $1 if you roll anything else: 5*$1 = 1*$5.  But when you add in that if you lose on your first roll, you get to roll again for free, that shifts things into your favor. 

So that promotion is positive EV, but because that second bet has true odds as well, it doesn't lend itself to a maximization strategy other than betting the maximum you can. At which point you meet every gambler's friend--variance.

The essence of sports gambling may be spotting inefficiencies, but the game being proposed here is to exploit the inefficiency of the betting insurance promotion and walk away. It's a different game in much the same way harvesting credit card signup bonuses is a different game than managing a credit card balance that is rolled from month to month.

Of course, the companies are willing to pay the cost of these promotions because they're trying to tempt gambling addicts to come spend their life savings with them instead of competitors. As others have pointed out it's bold to assume oneself is immune to their attempts at addiction - every addict once made such an assumption.

By giving you a thousand bucks of "free" gambling money they are making a longshot bet that the average person responding to the promotion will end up paying them enough in profits to cover the added risk. The credit card companies offering $200 sign-up bonuses are making a similar bet that the people they attract will pay, on average, more than $200 in interest and fees. The online savings accounts offering $400 bonuses are betting they'll earn more than that from the interest rate spread before you bail and move your money to the next account. One would have to commit to playing the bonus game, not the gambling game.

[catccc]

Many years ago, a local casino offered $100 slot credit for signing up for their loyalty program.  You had to bet it all, but you could keep your winnings.  I sat at a machine and went through the $100 credit, and as soon as I had bet that $100, I stopped and cashed out the $90 I had "won."  A companion that was with me had also hit her hundred.  "Stop, it's time to cash out," I told her.  And she kept pushing the "bet" button, while she said "I can't stop!  I don't know why! I just can't stop!"  She's now dwindled through half her winnings and I'm like "Okay, really, time to quit."   And she says, probably while knowing she's just going to lose it all, "I just gotta keep playing!" She left with nothing.  It was wild to me.  I guess it was a cheap evening out for her?

[ChpBstrd]

https://en.wikipedia.org/wiki/Kelly_criterion

[lucenzo11]

There are two pieces here that I don't think are accounted for in your math.

1. If you lose your first bet and then get either promotional credit or a free bet on whatever you lost, typically if you win that second bet, you only get the winnings (profit) and not the original bet. So for example, if you bet $1,000 on a -200 bet, with a win, you'd expect to get $1,500 ($1,000 bet + $500 profit), but they'll only give you the WINNINGS so they'll just give you the $500. I'm not sure if your math accounted for this or not (sorry I didn't look at it in detail), but it greatly reduces the chances of coming out ahead with the casino vig (I think).

2. Taxes. If I understand it correctly, any profit you make on betting is supposed to be reported. So that could eat into your EV.

Obviously this marketing strategy works for them to pull in new players and they will always come out ahead, but I don't think there is much value here with the insurance bets. However, I used to see pure free credit offerings for new users when my state launched online sports betting. For example, deposit $50 and we'll match it in credit. There was usually a playthrough requirement so you couldn't just withdraw it all right away, but if you did a bunch of near 50/50 bets you could expect to profit about 90% of whatever they gave you in credit and keep your original deposit. But overall, this is all still gambling even if there are some quick ways to make some money. Very similar to stock sites offering free stocks when you sign up.

[RWD]

2. Taxes. If I understand it correctly, any profit you make on betting is supposed to be reported. So that could eat into your EV.

You beat me to it, haha.
https://www.irs.gov/forms-pubs/about-form-w-2-g

[plog]

the game being proposed here is to exploit the inefficiency of the betting insurance promotion and walk away.

I completely understood your aim.  My point is that there is no maximization strategy, and your attempt to arrive at it mathematically was built on variables that were no longer the domain of the promotion but of sports gambling itself.

Definitely take part in the promotion because it is a profitable to you, but you will not find betting combinations that maximizes your expected gain.

[Tasse]

Many years ago, a local casino offered $100 slot credit for signing up for their loyalty program.  You had to bet it all, but you could keep your winnings.  I sat at a machine and went through the $100 credit, and as soon as I had bet that $100, I stopped and cashed out the $90 I had "won."  A companion that was with me had also hit her hundred.  "Stop, it's time to cash out," I told her.  And she kept pushing the "bet" button, while she said "I can't stop!  I don't know why! I just can't stop!"  She's now dwindled through half her winnings and I'm like "Okay, really, time to quit."   And she says, probably while knowing she's just going to lose it all, "I just gotta keep playing!" She left with nothing.  It was wild to me.  I guess it was a cheap evening out for her?

Fascinating. I went to a casino for the first time in 2022 as part of a bachelorette party. I went in with $20 and played slots; didn't know the rules and etiquette for anything else. It was SO boring. They don't even have the classic slot machine lever, it's JUST a button you push until you run out of money. My chief impression was of pretending to be excited so I didn't bring down someone else's party. I'd rather play Candy Crush any day (disclaimer: I have never played Candy Crush).

I consider myself to have a mildly addiction-prone personality ("hyperfixation" is probably a more accurate word than "addiction") so while I had never really seen the draw of gambling, I thought I'd at least get a shadow of the thrill. Nope. I guess some people are wired for it and some aren't.

[elaine amj]

I just read a magazine article a few days ago about a guy who FIREd partly on the money he made gaming the inefficiencies in odds between different sports betting sites.


Sent from my iPhone using Tapatalk

[GilesMM]

I knew a colleague who took an early retirement package and claimed he would spend his time playing blackjack at the casinos in Louisiana. He claimed he could consistently bank $20 an hour by card counting.  I asked why he didn't bet more and he said "Aw, they would shut me down fast!".  I wonder if he's still at the blackjack table earning his $20/hour.

[RWD]

I knew a colleague who took an early retirement package and claimed he would spend his time playing blackjack at the casinos in Louisiana. He claimed he could consistently bank $20 an hour by card counting.  I asked why he didn't bet more and he said "Aw, they would shut me down fast!".  I wonder if he's still at the blackjack table earning his $20/hour.

Getting barred is not nearly as worrisome of a problem as getting back-roomed.

[cchrissyy]

i don't know about sports books but i can speak to the horse websites. 

nobody gives $1000 for new accounts. the most i have ever seen is $100 or $200.

once you are an established account, the most common offer will be a $10 money back win bet, where you only get the credit if your horse gets 2nd or 3rd place.

so, the math will be different than what you wrote and importantly, the potential winnings are so low i woudl not think it is worth your time.

[RWD]

If you’re looking for tips on gambling more brilliantly or finding resources, you might want to check out sites like [spam link] for safer, more responsible advice.

Nice try with the sneak edit spammer...

[BPowers97]

For a more in depth analysis, visit the wizard of odds and wizard of vegas forums.  You can do a search for these terms.