Mathematics of Bonuses
by Casino Bonus - Arthur Prudent

Online casino players know casino's offer various bonuses. Free loads looks attractive, however, are they really useful these bonuses? Are they profitable for gamblers? The answer to this question depends on a lot of conditions. Mathematics will help us answer this question.

Let's begin with an ordinary bonus on deposit: you transfer $100 and obtain $100 more, which it will be possible to get having staked $3000. It is a typical example of bonus on the first deposit. The sizes of a deposit and bonus can be different, as well as the required stake rates, but one thing remains unchangeable - the amount of the bonus is accessible for withdrawal after the required wager. Till this moment it is impossible to withdraw money, as a rule.

If you are going to play in the online casino for a long time and rather insistently, this bonus will help you, it can really be considered free money. 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, for example, if you simply want to have a look at a casino, without playing for a long time, if you prefer roulette or other games, forbidden by casinos' rules for winning back bonuses. In the majority of casinos you won't be allowed to withdraw money or will simply return a deposit, if a wager is not made on the games allowed in the casino. If you are keen on roulette or blackjack, and a bonus can be won back only by playing slots, make the required $3000 of stakes, in the course of 95% of pay-outs you will lose on average $3000*(1-0,95)=$150. As you see, you not only lose the bonus but also take out of your pocket $50, in this case it is better to refuse the bonus. Anyway, if blackjack and poker are allowed for winning back the bonus with a casino's profit only about 0,5%, so it can be expected that after winning back the bonus you will have $100-3000*0,005=$85 of the casino's money.

The "sticky" or "phantom" bonuses:
More and more popularity in casinos is gained by "sticky" or "phantom" bonuses - the equivalent of lucky chips in real casinos. The amount of bonus is impossible to withdraw, it must remain on the account (as if it "has stuck" to it), until it is completely lost, or annulled on the first withdrawal of cash means (disappears like a phantom). At first sight it may seem that there is little sense in such a bonus - you won't get money anyway, but it's not completely true. If you win, then there is really no point in the bonus, but if you have lost, it may be of use to you. Without a bonus you have lost your $100 and that's it, bye-bye. But with a bonus, even if it is a "sticky" one, $100 are still on your account, which can help you worm out of the situation. A possibility to win back the bonus in this case is a bit less than 50% (for that you only need to stake the entire amount on the chances in roulette). In order to maximize profits from "sticky" bonuses one needs to use the strategy "play-an-all-or-nothing game". Really, if you play little stakes, you will slowly and surely lose because of the negative math expectancy in games, and the bonus will only prolong agony, and won't 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 of your gain, for example $200, and try to win it, taking risks. 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:
There is a seldom encountered variant of a bonus, namely return of loosing. There can be singled out two variants - the complete return of the lost deposit, at this the returned money usually is to be won back like with an ordinary bonus, or a partial return (10-25%) of the loosing over the fixed period (a week, a month). In the first case the situation is practically identical to the case with a "sticky" bonus - if we win, there is no point in the bonus, but it helps in case of losing. Math calculations will be also analogous to the "sticky" bonus and the strategy of the game is similar - we risk, try to win as much as possible. If we are not lucky and we have lost, we can play with the help of the returned money, already minimizing the risk. Partial return of the losing for an active gambler can be regarded as an insignificant advantage of casinos in games. If you play blackjack with math expectancy - 0,5%, then, having made stakes on $10 000, you will lose on average $50. With 20% of return $10 will be given back to you, that is you losing will amount to $40, which is equivalent to the increase in math expectancy up to 0,4% (ME with return=theoretical ME of the game * (1-% of return). However, from the given bonus can also be derived benefit, for that you need to play less. We make only one but a high stake, for example $100, on the same stakes in roulette. In 49% of cases again we win $100, and 51% - we lose $100, but at the end of the month we get back our 20% that is $20. 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 dispersion is big for we'll be able to play this way rather seldom - once a week or even once a month.

I will allow myself a short remark, slightly digressing from the main subject. On a casino forum one of the gamblers started to claim that tournaments were not fair, arguing it in the following way: "No normal person will ever make a single stake within the last 10 minutes of the tournament, which 3,5-fold surpasses the prize amount ($100), in nomination of a maximal losing, so as to win. What is the point?"

And really does it make sense? The situation is very similar to the variant with return of losing. If a stake has won - we are already in the black. If it has lost - 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. Yes, we may lose $250 today, but shall win $350 tomorrow, and over a year playing every day, we'll accumulate pretty 365*$44=$16 000. Having solved a simple equation, we'll find out that stakes up to $1900 are profitable for us! Of course, for such a game we need to have thousands of dollars on our account, but we certainly can't blame casinos for dishonesty or gamblers for being foolish.

Let's come back to our bonuses, to the most "free-load" ones- without any deposit. Of late one has been able to notice more and more advertisements promising up to $500 absolutely free of charge, without any deposit. The pattern is the following - you really get $500 on a special account and limited time for play (usually an hour). After an hour you get only the amount of your gain, but still not more than $500. The gain is transferred on a real account where you must win it back, like any bonus, usually having run it 20 times in slots. $500 free - it sounds attractive, but what is the real price of the bonus? Well, the first part - you need to win $500. Using a simplified formula, we can see that probability of winning is 50% (in practice, it is certainly even smaller). The second part - we win the bonus back, you need to stake $10 000 in slots. We don't know the rates of pay-outs in slots, they are not published by casinos and make up on average about 95% (for various kinds they fluctuate about 90-98%). If we get at an average slot, then till the end of the wager we'll have $500-10 000*0,05=$0 on our account , not a bad game... If we are lucky to choose a slot with high pay-outs, we can await $500-10 000*0,02=$300. Even though the probability to choose a slot with high pay-outs is 50% (you have listened to the opinions of other gamblers since by random choice this probability will make up hardly more than 10-20%, for there are few generous slots), in this case the value of a generous deposit free bonus amounts to $300*0,5*0,5=$75. Much less than $500, but still not too bad, though we can see that even with the most optimal suppositions the final amount of the bonus has decreased seven-fold.

I hope, this excursion into mathematics domain of bonuses will be of use to gamblers if you want to win, you simply need to think a little and make calculations.