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Online Casinos: The Mathematical Basis of Bonuses
Online casino players are well aware that these bonuses are offered at many casinos. "Free-load" is appealing, but are they really worth these bonuses? Are they lucrative for gamblers This is a question that depends on a variety of factors. The answer to this question is possible by math.

Let's start with a typical bonus when you deposit $100, and then receive another $100, which it will be possible to receive after you have put up $3000. This is an example of a bonus you receive on the first deposit. The size of the bonus and deposit can be different in addition to the stake rate required However, one thing remains in place - the amount of bonus is accessible for withdrawal after the wager is completed. At present, it's impossible to withdraw funds generally.

It is considered free money if you are playing at the casino online for a lengthy duration 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. But there can be complications, for example, if you want to simply take an experience at a casino without having to play for long or if you like roulette or other gamesthat are forbidden by casinos' rules for winning back bonuses. If you aren't betting in any of the permissible games, most casinos won't allow you to withdraw money. You can win a bonus by playing blackjack or roulette however, only if make the required 3000 stakes. If you're lucky enough to win 95% payouts, you'll lose an average of $3000* (1-0,95) = $150. As you see, you do not just lose the bonus but will also be able to take from your pocket $50, in the case of this, it's better to not accept the bonus. If you could win back the bonus by earning a profit of 0.5 percent, it's likely that you will get $100-3000*0,005=$85 after you have won back the bonus.
"sticky" or "phantom" bonus:

A growing amount of popularity in casinos is due to "sticky" or "phantom" bonuses, which are equivalent to lucky chips in real casinos. The amount of the bonus is impossible to withdraw and must stay in 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 an illusion). It might appear as if a bonus is not worthwhile. You will not be able to take any money out, but it's not the case. The bonus won't be worth the cost if you win. However, if you fail, the bonus might prove useful. You've already lost $100 with no bonus. If the bonus was not "sticky" it remains on your account. This will allow you to wiggle out of the situation. There is a chance to win back the bonus is less than 50 percent (for you will only have 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". In reality, if you are playing with small stakes, you'll slowly and surely lose due to the negative math expectation in games, and the bonus is likely to 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 suggested to set the desired amount you wish to profit, for instance $200, and try to win it, 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).

The cash back bonus:

It is not often seen variant of a bonus, namely return of losing. Two types of bonuses can be distinguished: the complete refund of deposit. At this point the deposit is generally won back just like an normal bonus. Or a partial return (10-25 percent) for a set period (a month or a week). The first scenario is almost similar to a "sticky bonus" The bonus is not worth it in the event of winning, but helps if you lose. In the second case, the "sticky bonus" mathematical calculation will be similar. The method of play for the game is the same - we gamble to win as frequently as possible. We can still gamble with the money you've earned even if we do not succeed. A partial return on the loss for a gambler who is active can be regarded as an insignificant benefit of casinos when playing games. You'll lose an average of $50 if you play blackjack with an average math expectation of 0.5%. If you earn 20% of the money, the amount of $10 is returned to you, which means that the amount your loss will be $40, which is equivalent to an increase in the math expectation up to 0,4 percent (ME with return=theoretical ME the game (1percent of return). But, from the bonus, you can also gain benefits, which means you'll need to play less. On the same stakes as on roulette, we play one, however it's an enormous stake. We win $100 in 49% of the cases however $100 is taken home by 51% of players. But, we have to lose $100 in 51% of the cases. At the end of each month, we get back 20% of our $20 winnings. As a result the effect is $100*0,49-($100-$20)*0,51=$8,2. The stake has a positive math expectation. The dispersion however is high and we'll only be able to play in this way for a few times per week or once per month.

Allow me to provide a short remark. This is a bit off topic. One forum member said that tournaments were unfair. He claimed, "No normal person will ever put a stake in in the final 10 minutes." The 3,5-fold increase is more than the prize amount ($100) in the nomination of maximum loss, meaning as not to lose. What's the reason?

It is logical. games to play is identical to that of the return of losing. 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 lose $250 today but earn $350 in the future. Over a year of playing every day and earning a total of 365, our earnings are quite impressive at 365*$44=$16 000. We'll see that stakes as high as $1900 are profitable for us if we solve an easy equation. It is essential to have many thousands of dollars in our accounts to play this game, however we can't blame the casinos for being dishonest or foolish.


Let's get back to our bonuses. They are among the most "free-loading" bonuses that do not require a deposit. Of late one has seen an increasing number of ads promising the possibility of up to $500 completely for free, and with no deposit. The way to look at it is this - you really get $500 with a separate account and limited time for playing (usually an hour). After an hour, you will receive just the amount of your gains, but not greater than $500. The cash is transferred to a real account where you have to be able to win it back, just like any bonus, usually having played it at least 20 times through slots. The $500 bonus sounds tempting but what's the actual value of this bonus? The first aspect is that you must win $500. Using a simplified formula, we will see that the probability of winning is 50% (in reality, it's certainly even smaller). To win the bonus back, you must stake 10 000 dollars on slot machines. We don't know the rates of payouts on slots, they are not provided by casinos, and average around 95% (for various kinds they fluctuate about 90-98%). An average slot will give us $500-10 000*0.05=$0. That's not an unreasonable amount. You can anticipate $500 to 000*0.02=$300 if we're lucky enough to land a lucrative slot. The likelihood of picking one with the highest payout is 50 percent. But, you've read the opinions of other gamblers , as this probability is not more than 10-20 10%. In this case the deposit bonus is generous of $300*0.5*0.5=$75. Although it is less than $500, this is still an excellent amount. However, we can observe that the bonus's total value has decreased sevenfold, even with the best possible estimations.

I'm hoping that this journey into mathematics domain of bonuses will be helpful to gamblers . If you are looking to be successful, you only must think about it and make calculations.

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