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Online Casinos: The Mathematics of Bonuses
Casino players who play online are aware that these bonuses are available in many casinos. "Free-load" appears attractive, however, are they really worth are they really worth it? Are they worth the money for gamblers? The answer to this question will depend on a lot of conditions. Mathematics will assist us in answering this question.

Let's start with a typical bonus on deposit: you transfer $100 and obtain $100 more and it's possible to get having put up $3000. This is an example of bonus for the first deposit. Although the size of a bonus or deposit may differ, so can the stake rate. However, look at here is certain: the bonus amount is still able to be withdrawn following the wagering requirement. As a rule, it is not possible to withdraw money.


This bonus can be considered free money if you gamble online for a lengthy period of time and you are consistent. 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 some complications. In particular when your objective is to simply have an overview of the casino without spending a lot of time there, or you like roulette or other games that are prohibited under bonus rules, you might be denied access to the bonus amount. In most casinos there is no way to withdraw money or will just return your deposit if a wager is not made on the games allowed in the casino. If you are keen on blackjack or roulette and you can returned only through playing slots, place the required $3000 of stakes and in the 95% payouts, you'll lose $3000*(1-0,95)=$150. cool games to play will are not just losing the bonus but will also be able to take from your account $50, in this scenario, it's best to not accept the bonus. If blackjack and poker can be used to claim back the bonus with a casino's earnings of only 0,5%, so it can be expected that after reclaiming the bonus you will have $100-$3000 plus 0,005 = $85 from the casino's profit.
"Sticky" as well as "phantom” bonuses

The popularity of casinos is derived from "sticky" or "phantom" bonuses - similar to luck chips in real casinos. The amount of the bonus cannot be withdrawn the bonus, and it will remain on the account (as as if it "has stuck" to it) until it's totally lost or canceled after the first time you withdraw cash means (disappears like it's a phantom). It might appear as if an offer isn't worth the effort. You won't be able to withdraw any money, but this isn't the case. If you win, there's no reason to the bonus. However, even if you lose it might help you. Without a bonus you have lost your $100 and that's it, bye-bye. With a bonus, even if it is an "sticky" one, you will find that $100 are still on your account, which can help you worm out of the situation. The chance of winning back the bonus in this case is less than 50% (for that you only need to bet the whole amount on the chance of winning in roulette). In order to maximize profits from "sticky" bonuses one needs to use the strategy "play-an-all-or-nothing game". In reality, if you are playing with low stakes, you'll slowly and surely lose because of the negative mathematical expectation in games, and the bonus is likely to prolong suffering, and won't aid you to 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. You should set the amount you wish to earn, like $200, and take risks to be successful. 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).

Cash back Bonus:

The most common bonus seen is the return of lost funds. Two types of bonuses could be distinguished: the full refund of deposit. In this case, the deposit is generally returned just as an ordinary bonus. A partial return (10-25 percent) for a set period (a month or a week). In the second case, the situation is practically identical to that of the "sticky" bonus - if we win, there is no need for the bonus, however, it helps in case of losing. Math calculations will be also analogous to "sticky" bonus, and the strategy is the same: we take risks, try to win as many times as we can. We can still play with the money we've won, even if we do not take home the prize. Casinos in games can offer a partial return on losing for gamblers who are active. You will lose $50 on average playing blackjack with a math expectancy of 0.5%. The payout is $10 when you lose $20. This is equal to the math expectancy rise of 0.4 percent. But, from the bonus can also be derived from the fact that you'll need to play less. You only make one, however very high stake, like $100, using the same roulette stakes. The majority of the cases we again win $100, and 51% - we lose $100, but at the end of the month we get back our 20%, which is 20 dollars. As a result the effect is $100*0,49-($100-$20)*0,51=$8,2. As you see, the stake then has positive math expectancy, but its dispersion is huge, since you to play in this manner very rarely - at least once per week or once per month.

slot game 'll let myself make to make a brief remark, but slight deviation from the main issue. In a forum for casinos, one gambler began to assert that tournaments were not fair, arguing it in the following way: "No normal person will ever be able to make a single wager in the final 10 minutes of the tournament and this is 3,5-fold more than the prize ($100) and in the event of a loss that is as high as so that they can take home a prize. What is the point?"

And really does it make sense? It's identical to the scenario that has a return on losing. If a stake is successful - we are already in the black. If it loses - we'll get a tournament prize of $100. So, the math expectancy of the above-mentioned stake amounting to $350 is: $350*0,49-($350-$100)*0,51=$44. Sure, we could lose $250 today, but we will win $350 in the future. Over a year of playing each day the total amount we earn are quite impressive , at 365*$44=$16000. If we can solve a basic equation, we'll find out that stakes up to $1900 are profitable for us! It is essential to have several thousand dollars on our accounts for this game, however we shouldn't blame casinos for being shady or naive.

Let's go back to our bonuses. They are among the most "free-loading" bonuses that do not require a deposit. Recently, one has seen an increasing number of ads promising as much as $500 free of charge, without any deposit. You will receive $500 in exchange for a special account, and a limited amount of time to play (usually one hour). After an hour you get only the amount you gain, but still not greater than $500. The cash is transferred to a real account where you are required to be able to win it back, just like any other bonus, generally after having played it at least 20 times in slot machines. This sounds fantastic but what's the exact cost of this bonus? Well, the first part requires you to be able to win $500. It is evident that the probability of winning $500 is 50% based on an easy formula. However, in practice it is much lower. To win the bonus back, you must stake 10 000 dollars on slot machines. We don't know the rates of pay-outs in slot machines, but they are published by casinos and average around 95 percent (for various kinds they fluctuate around 90-98 percent). An average slot will give us between $500 and 000*0.05=$0. This isn't an unreasonable amount. We can expect $500-10 000*0.02=$300 If we're fortunate enough to locate a high-paying slot. Even though the likelihood to choose a slot with high pay-outs is 50% (you are probably familiar with the opinions of other gamblers since the random selection of this probability will be less than 10-20%, for there are a few slots that pay out generously) In this instance, the worth of a large deposit free bonus amounts to $300*0,5*0.5%=$75. Even though it's not $500, it's still an impressive amount. However, we are able to observe that the bonus's total value has decreased sevenfold, even with the best estimations.

I hope this exploration into the mathematical realm of bonuses can prove beneficial to gamblers. If you want to be successful, all you need is to think and make calculations.

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