Notes

Online Casinos: Mathematics of Bonuses
Casino players who play online are aware that these bonuses can be found in many casinos. While "Free-load" might seem appealing, they are not really worth the effort. Are they profitable for gamblers? This is a question that depends on a variety of factors. Mathematics will help us answer this question.

Let's begin with the typical bonus on deposit. You deposit $100 and receive another $100. This is possible after you have staked $3000. This is an instance of a bonus on your first deposit. The amount of deposit and bonus can be different in addition to the required stake rates However, one thing remains in place - the amount of bonus is available for withdrawal after the wager is completed. Till this moment it is impossible to withdraw funds generally.

It is considered free money when you gamble online for a long duration and keep playing. If you play slots with 95% pay-outs, a bonus will allow you to make on average extra 2000 $ of stakes ($100/(1-0,95)=$2000), after that the amount of bonus will be over. There are a few pitfalls. For example, if your goal is to just take a look at the casino, without spending a lot of time there, or if you enjoy roulette or other games that are prohibited by bonus rules, you might be denied access to the bonus. In the majority of casinos you won't be allowed to withdraw cash or just return your deposit when a wager isn't made on games permitted in the casino. If you're a fan of blackjack or roulette, and a bonus can be won back only by playing slot machines, you must make the minimum stakes of $3000 and in the 95% payouts, you will lose on average $3000*(1-0,95)=$150. You will lose $50, and forfeit the bonus. In this scenario it's better not to accept the bonus. In any case, if blackjack and poker are allowed to claim back the bonus with a casino's profits of just 0,5%, so you can expect that after winning back the bonus you will have $100-$3000 plus 0,005 = $85 from the casino's profit.
"Sticky" as well as "phantom” bonuses


A growing amount of popularity in casinos is gained by "sticky" or "phantom" bonuses - equivalent to casino chips that are lucky in real life. The bonus amount is impossible to withdraw, it must remain on the account (as when it "has been glued" to it), until it is totally lost or canceled after the first time you withdraw cash means (disappears like it's a phantom). It could appear that an offer isn't worth the effort. It isn't possible to withdraw any money, but this isn't the case. The bonus won't be worth it if you are successful. However, if you lose, it might prove useful. Already, you've lost $100 with no bonus. However, with a bonus even if it is one that is "sticky" one, you will find that $100 are still on your account, which can help you worm out of the circumstance. The chance of winning back the amount you received is less than half (for this you will only have to put the full amount on roulette). In order to maximize profits from "sticky" bonuses one needs to use the strategy "play-an-all-or-nothing game". If you only play small stakes, you'll slowly and surely lose because of the negative mathematical expectation in games, and bonuses will only add the pain, and will not help you win. Clever gamblers usually try to realize their bonuses quickly - somebody stakes the entire amount on chances, in the hope to double it (just imagine, you stake all $200 on chances, with a probability of 49% you'll win neat $200, with a probability of 51% you'll lose your $100 and $100 of the bonus, that is to say, a stake has positive math expectancy for you $200*0,49-$100*0,51=$47), some people use progressive strategies of Martingale type. It is recommended to fix the desired amount you wish to profit, for instance $200, and then try to win it by taking risk. If you have contributed a deposit in the amount of $100, obtained "sticky" $150 and plan to enlarge the sum on your account up to $500 (that is to win $250), then a probability to achieve your aim is (100+150)/500=50%, at this the desired real value of the bonus for you is (100+150)/500*(500-150)-100=$75 (you can substitute it for your own figures, but, please, take into account that the formulas are given for games with zero math expectancy, in real games the results will be lower).

The cash back bonus:

One bonus that is seldom seen is the return of lost. Two kinds of bonuses can be distinguished from the total return of the deposit. The deposit is generally returned as an ordinary bonus. A partial return (10-25%) for a set period (a week or a month). The first scenario is nearly the same as a "sticky bonus" The bonus will not be worth anything in the event of winning, but helps when you lose. Math calculations are similar to "sticky" bonus and the strategy is similar - we risk and try to win as many times as we can. You can play with the money you've earned even if we don't win. Casinos with games offer a partial return on losing for gamblers who are active. You will lose about $50 playing blackjack with an average math expectation of 0.5 percent. If you earn 20% of the money, the amount of $10 is returned to you. That means your loss will be $40, which is equivalent to the increase in math expectancy to 0,4% (ME with return=theoretical ME the game * (1percent of return). It is possible to still profit from the bonus but you'll have to be playing less. There is only one bet, but a high stake, for example $100, with the same roulette stakes. The majority of the cases we also win $100 and 51% - we lose $100, but at the end of the month we win back 20%, which is 20 dollars. As a result the effect is $100*0,49-($100-$20)*0,51=$8,2. It is evident that the stake then has positive math expectation, however the it's a big dispersion, as it to be played this way rather seldom - once a week or even once a month.

I will allow myself a short remark, slight deviation from the main issue. One of the forum members declared that tournaments weren't fair. He said, "No normal person will ever put a stake in within the final 10 minutes." The 3,5-fold increase is more than the amount of prize ($100) in the case of the maximum loss, so that you won't lose. What's the reason?

It makes sense. The situation is quite similar to the one with return of losing. The stake is in the black if the stake has been taken home. We'll receive a tournament prize of $100 in the event that it fails to win. So, the math expectancy of the above-mentioned stake amounting to $350 is: $350*0,49-($350-$100)*0,51=$44. Yes, we might lose $250 today but earn $350 in the future. In the course of a year of daily play the total amount we earn will be quite impressive , at 365*$44=$16000. It's clear that stakes of up to $1900 could be profitable for us if we solve an easy equation. It is essential to have thousands on our accounts for this kind of game, but we don't have to blame casinos for being untruthful or inexperienced.

Let's look back at our bonus offers, especially the most "free-load" ones- without any deposit. Recently, one has been able to notice increasing numbers of advertisements that promise the possibility of up to $500 completely free of charge, without any deposit. The pattern is the following you actually get $500 with a separate account, and a limited amount of time to playing (usually an hour). You will only get the winnings after one hour, but never more than $500. You have to win the bonus back on a real account. In click here for more , you've run it 20 times in slot machines. $500 for free sounds appealing however, what is the actual value of this bonus? The first part is that you need to win $500. Using a simplified formula, we can see the odds of winning are 50% (in the real world, it's definitely lower). If you want to get the bonus back and you have to bet $10 000 in slots. The pay-out percentages of slot machines are not known. They average around 95% and fluctuate between 90-98% for different types. If we play an average slot, at the end of our wager , we'll be able to deposit $500-10 000*0,05=$ in our bank account, which is not an excellent game... If we are lucky to choose a slot with large payouts, we can look forward to $500-10 000*0,02=$300. Even though the probability to choose a slot with high pay-outs is 50 percent (you have listened to the comments of other gamblers as the random selection of this probability will make up hardly more than 10-20%, as there are only a handful of slots with high payouts) In this instance, the value of a huge deposit free bonus amounts to $300*0,5*0.5%=$75. Although it is less than $500, this is still an impressive amount. But, we can see that the bonus's final value has dropped sevenfold even with the best possible assumptions.

I hope, this excursion into mathematics domain of bonus will prove useful to gamblers - if you're looking to win, you just need to think a little and calculate.

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