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Online Casinos: The Mathematical Basis of Bonuses
Online casino players are well aware that these bonuses are offered in many casinos. While "Free-load" might seem appealing, they are not worth the effort. Are they profitable for gamblers? The answer to this question is contingent on many factors. Mathematical knowledge can aid us in answering this question.

Let's begin with the typical bonus when you deposit. The deposit is $100, and you receive another $100. This is feasible after you stake 3000. It is a typical instance of a bonus for the first deposit. The size of the bonus and deposit can be different, as well as the stake requirements However, one thing remains unchangeable - the amount of bonus can be withdrawn after the required wager. Till this moment it is impossible to withdraw funds generally.

If you intend to play at the online casino for a long period of time, and you are persistent about it the bonus can 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 such as if you want to simply take the chance to play at a casino without playing for a long time or if you like roulette or other gamesthat are which are not permitted by the rules of casinos for winning back bonus. In most casinos there is no way to withdraw funds or simply return a deposit, if a wager is not made on games permitted in the casino. If you're keen on roulette or blackjack, and a bonus can be returned only through playing slots, place the required $3000 of stakes during paying out 95% of the time you will lose on average $3000*(1-0,95)=$150. You will not only lose the bonus, but you will also be able to take from your account $50, in the case of this, it's better to decline the bonus. If blackjack and poker can be used to claim back the bonus and the casino's earnings of only 0,5%, then you can expect that after winning back the bonus, you'll be left with $100-$3000, which is a total of 0,005 or $85 for.
"sticky" or "phantom" bonus:

Casinos are becoming increasingly popular because of "sticky" and "phantom bonuses. These bonuses are equivalent to the lucky chips found in a real casino. It isn't possible to cash out the bonus. The bonus amount must be placed on the account as if it "has stuck". 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 is not worth it if you are successful. But, if you lose, it may be beneficial. You have already lost $100, without a bonus. Even if the bonus was not "sticky" it will still be in your account. This can help to get out of the situation. The chance of winning back the bonus is a bit less than 50% (for it is only necessary to stake the entire amount on the odds of 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 low stakes, you'll eventually lose money because of the negative mathematical expectations in games. Moreover, the bonus will only 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. It is important to determine the amount you would like 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).

The cash back bonus:

It is a rare variation of a bonus namely return of losing. It is possible to distinguish two variants - the complete return of the deposit that was lost, at this the returned cash is typically won back like with any other bonus, or a partial return (10-25 percentage) of the amount lost over the fixed period (a week, a month). The first situation is similar to the "sticky bonus" - the bonus is useless in the event of winning however it can be helpful when you lose. Math calculations are analogous to "sticky" bonus, and the strategy of the game is the same - we take risks, try to win as many times as possible. If we don't win and lose, we can play with the help of that money back, thus minimizing the risk. A partial return on the loss gambler could be seen as a minor benefit of casinos in games. If you are playing blackjack with math expectancy - 0,5%, then after you have staked $10,000, you'll lose an average of $50. You will receive $10 back if you make a loss of $20. This is equivalent to the math expectancy increase of 0.4 percent. You can still derive benefits from the bonus but you'll have to play less. With the same stakes in roulette, we place one, however it's the largest stake. The majority of the cases we again win $100, and 51% of the time we lose $100. However, at the end of the month we win back 20% which is equal to $20. As a result the effect is $100*0,49-($100-$20)*0,51=$8,2. It is evident that the stake is then positive in math probability, but the it's a big dispersion, as we'll be able to play in this manner very rarely - once a week or even every month.

I'll allow myself an unintentional remark that is slightly digressing from the main issue. One forum member claimed that tournaments weren't fair. He claimed, "No normal person will ever be able to stake a single stake during the final 10 minutes." free games to play is 3,5 times the prize amount ($100) in the nomination of maximum loss, meaning it's impossible to lose. What is the point?"

It makes sense. It's very similar to the variant that has a return on losing. If a stake is successful the stake is already in the black. The stake will be awarded a prize of $100 if the stake 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 may lose $250 today, but shall win $350 tomorrow and, over the course of a year playing every day, we'll earn $16,000. We'll see that stakes of up to $1900 are profitable after solving a simple equation. Of course, for such a game , we'll need to have thousands of dollars on our account and we can't accuse casinos of dishonesty or gamblers who are foolish.

Let's go back to our bonuses. They're the top "free-loading" bonuses that do not require a deposit. Of late one has been able to notice increasing numbers of advertisements that promise up to $500 absolutely free of charge, without deposit. You get $500 for an account with a specific number of players, as well as you have a time limit to play (usually one hour). You will only get the amount of your win after one hour, but never more than $500. The gain is transferred on an actual account and you have to win it back, like any other bonus, generally after having run it 20 times in slots. $500 free -it sounds attractive, but what is the exact cost of the reward? The first part is that you must be able to win $500. It is evident that the probability of winning $500 is 50% based on the simplified formula. In reality, it's much lower. The second part - we get the bonus back You must bet at least $10 000 on slots. The pay-out percentages of slot machines aren't well-known. They are generally around 95% and fluctuate between 90-98% for different types. An average slot will give us $500-10 000*0.05=$0. It's not an awful amount. You can anticipate $500 to 000*0.02=$300 If we're fortunate enough to locate a high-paying slot. The probability of choosing the slot that has high payouts is 50 percent. But, you've heard the opinions of other gamblers , as the probability of winning will be between 10-20%. In this instance, the generous deposit bonus of $300*0.5*0.5=$75. While it's less than $500, it's still an impressive amount. However, we can observe that the bonus's total value has decreased sevenfold, even with the best possible estimations.


I hope, this excursion into the mathematics of bonuses will be of use to gamblers - if you're looking to win, you simply have to think about it and do some calculations.

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