Notes

Online Casinos: The Mathematical Basis of Bonuses
Online casino players are aware that these casinos offer a range of bonuses. "Free-load" looks appealing, but do they actually provide such bonuses? Are they lucrative for gamblers This is a question that depends on a variety of factors. This question can be answered using math.

Let's begin with an ordinary bonus upon deposit: you make $100, and then receive another $100 and it's possible to receive after you have staked $3000. It is a typical instance of a bonus for the first deposit. Although the size of a bonus or deposit can vary as well as the stake rates. However, one thing is sure: the bonus amount can be taken out after the wagering requirement. It generally is impossible to withdraw any funds.


If you intend to play at the online casino for a long time and rather insistently the bonus can assist you. It can be considered as 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 in the event that you want to simply take a look at a casino without playing for a long period of time and you are a fan of roulette or other games, forbidden by casinos' rules to win back bonuses. In most casinos, there is no way to withdraw cash or simply return a deposit, if a wager is not placed on the games that are allowed in the casino. If you're a fan of blackjack or roulette and a bonus is returned only through playing slot machines, you must make the required $3000 of stakes, in the course of the 95% payouts, you'll lose an average of $3000*(1-0,95)=$150. As you see, you not only lose the bonus, but you also take out of your wallet $50. In this scenario, it's best to not accept the bonus. If you will be able to recoup the bonus, with a profits of 0.5 percent, it's possible that you will get $100-3000*0,005=$85 after you have won back the bonus.
"sticky" or "phantom" bonus:

The popularity of casinos is derived from "sticky" or "phantom" bonuses - the equivalent of luck chips in real casinos. The amount of bonus cannot be taken out and must stay in the account (as as if it "has been glued" to it) until it's entirely lost or is canceled after the first time you withdraw cash means (disappears like a phantom). On first glance, it might appear that there is no sense in such a bonus - you won't receive any money however this isn't true. The bonus won't be worth it if you are successful. However, if you fail, the bonus may be beneficial. Without a bonus you have lost $100, and then you're gone. With a bonus, even if it's an "sticky" one, you will find that $100 remain on your account, which could assist you in getting out of the situation. A possibility to win back the bonus in this case is around 50 percent (for that you only need to put 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". If you only play small stakes, you'll gradually and eventually lose because of the negative math expectancy in games, and the bonus is likely to prolong suffering, and won't aid 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 winnings, such as $200, and try to win it, while taking chances. 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 noticed is the return of the money that was lost. There can be singled out two variants - the complete return of the deposit that was lost and the amount is usually won back like with an ordinary bonus, or a partial return (10-25%) of the loss over the fixed period (a week, a month). The first scenario is nearly identical to that of the "sticky bonus" - the bonus is not worth it in the event of winning however, it is beneficial in the event that you lose. In the second case, the "sticky bonus" calculation of math will be analogous. The principle of the game is the same - we gamble to win as frequently as is possible. If games to play don't win and we have lost, we can play with the help of that money back, thus minimizing the risk. The partial refund of losses for an active gambler can be regarded as an insignificant benefit of casinos when playing games. If you play blackjack with math expectancy of 0,5%,, having made stakes on 10 000 dollars, you'll lose an average of $50. If you earn 20% of the money, $10 will be given back to you. That means your loss will be $40, which is equivalent to an increase in math expectancy up to 0,4 percent (ME with return=theoretical ME of the game (1-% of return). But, from the bonus you will also get from the fact that you'll need to play less. With the same stakes in roulette, we play one, however it's an enormous stake. The majority of the cases again we win $100, and 51% - we lose $100. However, at the close of the month, we receive our 20% which is equal to $20. As a result the effect is $100*0,49-($100-$20)*0,51=$8,2. The stake has a positive math expectancy. However, free play is high and we'll only be able to play in this way for a few times each week or every month.

I will allow myself to make a brief remark, but slightly digressing from the main topic. On a casino forum one of the gamblers began 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 a tournament and this is 3,5-fold more than the prize amount ($100) as a result of a maximal losing, so as to win. What's the point?

It is logical. The situation is identical to that of return on losing. We are in the black if the stake has been taken home. 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 might lose $250 today but win $350 next day. Over a year of playing every day and earning a total of 365, our earnings are quite amazing at 365*$44 = $16,000. We'll discover that stakes as high as $1900 could be profitable for us if we solve an easy equation. internet 'll need many thousands of dollars in our accounts to play this game, but we don't have to blame casinos for being untruthful or inexperienced.

Let's revisit our bonuses, to the highest "free-load" ones- with no requirement for any deposit. There are more and more ads promising $500 for free, with no deposit. The basic idea is as follows You actually receive $500 with a separate account, and a limited amount of time to play (usually an hour). After an hour, you will receive only the amount of your gains, but not more than $500. The money is transferred to an actual account, where you have to be able to win it back, just like any other bonus, generally after having run it 20 times through slot machines. It sounds great but what's the exact value of this bonus? Well, the first part is that you must be able to win $500. By using a simple formula, we will see the odds of winning are 50 percent (in practice, it is likely to be even lower). In order to receive the bonus, you need to stake 10 000 dollars in slot machines. The pay-out rates in slot machines are not known. They average around 95%, but can vary between 90-98% for different types. A typical slot can give us between $500 and 000*0.05=$0. That's not an unreasonable amount. If we happen to select a slot that has 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% (you are probably familiar with the opinions of other gamblers , since the random selection of this probability will be less than 10-20%, as there are only a handful of slots with high payouts) in this scenario, the worth of a large deposit free bonus amounts to $300*0,5*0,5=$75. Even though it's not $500, it is an impressive amount. But, we can see that the bonus's final value has decreased by sevenfold even with the best estimations.

I'm hoping this look into the mathematical realm of bonuses can prove beneficial to gamblers. If you want to win, all you need is to think and make calculations.

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