Online Casinos: The Mathematics of Bonuses Casino players who play online know that casinos that provide a variety of bonuses. "Free-load" is attractive, however, are they really useful such bonuses? Are they profitable for gamblers? This is a question that depends on many different factors. Mathematical knowledge can aid us in answering this question.

Let's begin with an ordinary bonus upon deposit: you make $100 and receive $100 more, which it will be possible to get having put up $3000. This is an example of a bonus earned on the first deposit. While the amount of a deposit or bonus may vary and so do the stake rate. However, there is one thing that is for sure: the bonus can still be withdrawn after the wagering requirement has been met. In general, it is impossible to withdraw money.

The bonus is as free money if you are playing at the casino online for a lengthy duration and are persistent. 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)=$2000), after that the amount of bonus will be over. But there can be complications in the event that you simply want to have the chance to play at a casino, without having to play for long or if you like roulette or any other game, prohibited by casino rules to win back bonuses. In most casinos, you won't be allowed to withdraw cash or simply refund a deposit when a wager isn't made on the games allowed at the casino. There is a chance to win a bonus when you play roulette or blackjack however only if you meet the minimum stakes of 3000. In business , you'll lose an average of 3000$ (1-0,95) = $150. You will are not just losing the bonus, but will also be able to take from your wallet $50. In this case it is better to not accept the bonus. Anyway, if blackjack and poker are allowed for winning back the bonus with a casino's profit only about 0.5%, it can be expected that once you have repaid the bonus you will have $100-$3000 plus 0,005 = $85 from the casino's profit.
"Sticky" and "phantom" bonuses

A growing amount of popularity in casinos is due to "sticky" or "phantom" bonuses, which are the equivalent of casino chips that are lucky in real life. It's not possible to withdraw the bonus amount. The bonus has to be stored on the account as if it "has been shackled". On virtual reality , it might appear that there is no value in a bonus - you won't receive any money, but it's not completely accurate. The bonus won't be worth the cost if you win. If you fail, the bonus may be useful. Without games to play at home have lost your $100 and you're done. With a bonus, even if it's a "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 amount you received is less than half (for this, you'll have to bet the entire amount on roulette). In order to maximize profits from "sticky" bonuses one needs to use the strategy "play-an-all-or-nothing game". It is possible to lose slowly but surely if you stake in small amounts. The math expectancy that is negative of games means that you will not receive any bonuses. 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 $2000,49-$1000,51=$47), some people use progressive strategies of Martingale type. You should set the amount you want to earn, like $200, and then take the risk to win it. 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:

One bonus that is seldom recognized is the possibility of returning money the money that was lost. Two types of bonuses can be distinguished from the total refund of deposit. The cash is typically won back just like an normal bonus. Also, a partial return (10-25%) for a set period (a month or a week). In the second case, the situation is practically identical as with a "sticky" bonus. In the event that we win, there is no point in the bonus, but it helps in case of losing. Calculations in math will also be similar to "sticky" bonus and the strategy of the game is similar - we risk trying to win as many times as we can. If we do not win and we have lost then we are able to play again using this money, thus decreasing the risk. Casinos with games offer a partial return on losing for gamblers who are active. It is possible to lose $50 on average if you play blackjack with an average math expectation of 0.5 percent. A 20% return $10 will be given back to you. That means the loss you'll suffer is 40 dollars, which is comparable to an increase in math expectancy to 0,4% (ME with return=theoretical ME the game (1- % of return). But, from the bonus you will also get benefits, which means you need to be playing less. You only make one, however an extremely high stake, like $100, using the same stakes in roulette. In 49% of cases we also win $100 and 51% - we lose $100. However, at the time the month is over, we get back our 20% that is $20. As a result the effect is $100
0,49-($100-$20)0,51=$8,2. The stake has a positive math probability. The dispersion however is huge and we'll only be able to play this way once every week or once a month.

Allow me to provide a short remark. I'm a little off-topic. In a forum about casinos, one gambler began to assert that tournaments were not fair, arguing it as follows: "No normal person will ever be able to make a single wager within the last 10 minutes of the event that is 3,5 times greater than the prize amount ($100) and in the event of a maximal losing, so as to win. What's the purpose?

It is logical. The scenario is very similar to the variant that involves losing a stake. If a stake has won it is already in the black. If it is lost, we'll be awarded a prize in a tournament of $100. So, the math expectancy of the above-mentioned stake amounting to $350 is: $350
0,49-($350-$100)0,51=$44. We could be losing $250 right now, however we we'll get $350 the next day and, over the course of a year playing every day, we'll earn 16 000 dollars. We'll discover that stakes up to $1900 could be profitable after solving the simplest equation. Of course, in order to win at this kind of game, we'll require many thousands of dollars in our accounts however, we shouldn't blame casinos for dishonesty or gamblers for being naive.

Let's look back at our bonuses, specifically the most "free-load" ones- with no requirement for any deposit. In recent times, we've been able to notice more and more advertisements promising up to $500 absolutely free of charge, without deposit. You can get $500 on an account with a specific number of players, as well as only a certain amount of time to play (usually one hour). The only thing you will get is the amount you win after an hour, but no more than $500. The money is transferred to a real account where you have to win it back, like any bonus, usually when you have played it 20 times through slots. This sounds fantastic however, what is the real value of the bonus? Well, the first part requires you to be able to win $500. Based on a simplified formula, we can see that probability of winning is 50% (in reality, it's likely to be even lower). In order to receive the bonus, you need to stake 10 000 dollars in slot machines. The payout rates of slot machines aren't known. They range from 95 to 95%, but can vary between 90-98% for various kinds of. If we choose an average slot, then till the end of the wager we'll have $500-10 000
0,05=$0 on our account, not an excellent game... You can anticipate $500 to 0000.02=$300 If we're fortunate enough to land a lucrative slot. Even though the probability to select a slot with high pay-outs is 50 percent (you have heard the opinions of other gamblers , since randomly, this chance is less than 10-20%, as there are a few slots that pay out generously) In this instance, the value of a generous deposit free bonus amounts to $3000.5%*0.5%=$75. A lot less than $500 however, it's still not poor, although we can find that even with most ideal assumptions, that the total value of the bonus has diminished seven times.

I'm sure this trip into the mathematics of bonus will prove of use to gamblers - if you want to win, you simply have to think about it and calculate.