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Online Casinos: The Mathematics of Bonuses
Online casino players are well aware that bonuses are offered at many casinos. While "Free-load" might sound over at this website , they are not worthwhile. Are they profitable for gamblers? The answer to this question depends on a variety of factors. This question can be answered with mathematics.

Let' best games to play with a normal bonus for deposits. You deposit $100 and receive another $100. This will be possible after you have staked $3000. This is an example of a bonus earned on your first deposit. The amount of bonus and deposit can be different, as well as the stake requirements, but one thing remains unchangeable - the amount of bonus is accessible for withdrawal after the wager is completed. Till this moment it is impossible to withdraw cash, as a rule.


The bonus is as free money if you play at the casino online for a long 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. There are a few pitfalls in the event that you want to simply take the chance to play at a casino, but not playing for a long time or if you like roulette or other games, which are not permitted by the rules of casinos to win back bonuses. In most casinos you won't be allowed to withdraw funds or simply refund a deposit if a wager is not placed on the games that are allowed at the casino. If you're keen on blackjack or roulette, and a bonus is won back only by playing slots, place the minimum stakes of $3000 during the 95% payouts, you'll lose an average of $3000*(1-0,95)=$150. You lose $50 and also forfeit the bonus. In this scenario it's best not to accept the bonus. If blackjack and poker are allowed to win back the bonus and the casino's earnings of only 0,5%, so you can expect that after reclaiming the bonus you will have $100-$3000 plus 0,005 = $85 from the casino's profit.
"Sticky" as well as "phantom” bonuses

Casinos are becoming more popular because of "sticky" and "phantom bonuses. These bonuses are the equivalent to lucky chips in real casinos. It is not possible to cash out the bonus. The bonus must be stored on the account as if it "has been shackled". On first glance, it might seem that there is little value in a bonus - you won't get money anyway however, this isn't correct. If you are a winner, there is really no point to the bonus. However, in the event that you lose the money, it could be of use to you. You have already lost $100 without a bonus. However, with a bonus even if it's a "sticky" one, you will find that $100 are still in your account, which could assist you in getting out of the situation. A possibility to win back the amount of bonus is a bit less than 50 percent (for that you only need 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". You will lose slowly and sure if you only stake tiny amounts. The negative math expectation of the game means you won't get any bonus. free play 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 advised to determine the amount you want to gain, for example $200, and attempt 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:

The most common bonus recognized is the possibility of returning money lost funds. Two types of bonuses can be distinguished: the full return of the deposit. In this case, the cash is typically returned just as an normal bonus. A partial return (10-25%) for a set period (a month or a week). In the first scenario, the scenario is essentially identical to the case with a "sticky" bonus - if we win, there's no point in the bonus, however, it can be helpful in the event of loss. Math calculations are similar to the "sticky" bonus, and the strategy of the game is similar - we risk and try to win as much as possible. If we are not lucky and lose then we are able to play again with the help of the returned money, already minimizing the risk. Casinos with games offer some kind of compensation for losses to gamblers who have a high level of activity. If you are playing blackjack with math expectancy of 0,5%, then after you have staked $10,000, you'll lose an average of $50. You will receive $10 back when you lose 20 dollars. This is equivalent to an increase in math expectancy of 0.4%. But, from the bonus you will also get from the fact that you'll need to play less. With the same stakes on roulette, we place one, but it's a large stake. We can win $100 in 49% of the cases however $100 is won by 51%. However, we lose $100 in 51% of the cases. When we finish each month, we get back 20 percent of the $20 we won. As a result the effect is $100*0,49-($100-$20)*0,51=$8,2. As you see, the stake has a positive math expectation, however the its dispersion is huge, since we'll be able to play in this manner very rarely - at least once per week or once a month.

I'd like to make a brief remark. This is a bit off topic. On a casino forum one of the gamblers began to argue that tournaments are unfair. They argued in the following way: "No normal person will ever make a single stake within the last 10 minutes of a tournament, which 3,5-fold surpasses the amount of prize ($100) as a result of a maximal losing, in order to win. What's the purpose?"

It makes sense. It's like the one that has loss of money. If a stake has been won the stake is already in the black. The stake will be awarded a prize of $100 in the event that it loses. So, the math expectancy of the above-mentioned stake amounting to $350 is: $350*0,49-($350-$100)*0,51=$44. Sure, we could be losing $250 right now, however we we'll be able to win $350 next day and, over the course of a year playing each day, we'll build up $16,000. We'll see that stakes as high as $1900 can be profitable for us if we solve the simplest equation. Of course, for this kind of game, we'll need to have thousands of dollars on our accounts and we can't accuse casinos of dishonesty or gamblers for being naive.

Let's come back to our bonuses, to the best "free-load" ones- with no requirement for any deposit. Recently, one has noticed increasing numbers of advertisements that promise the possibility of up to $500 completely for free, and without any deposit. The basic idea is as follows - you really get $500 with a separate account, and a limited amount of time to playing (usually an hour). You'll only receive the amount of your win after an hour, but no more than $500. The bonus must be redeemed back on a real account. In most cases, you've run it 20 times in slot machines. The $500 bonus sounds tempting however, what is the actual value of the reward? The first thing to consider is is that you must be able to win $500. Based on technology simplified formula, we will see that probability of winning is 50% (in reality, it's certainly even smaller). To get the bonus back it is necessary to bet $10 000 on slots. The payout rates of slot machines aren't known. They are generally around 95%, and can range from 90-98% for different types. If we choose an average slot till the end of the wager we'll have $500-10 000*0,05=$0 in our bank account, which is not an awful game... We can expect $500-10 000*0.02=$300 If we're fortunate enough to locate a high-paying slot. Even though the probability to choose a slot with high pay-outs is 50% (you have listened to the opinions of other gamblers , since randomly, this chance will make up hardly more than 10-20%, for there are a few slots that pay out generously) In this instance, the worth of a large deposit bonus is $300*0,5*0.5%=$75. It's less than $500 but still not too poor, although we can observe that even with the most ideal assumptions, the value of the bonus has decreased seven-fold.

I am hoping that this investigation into the mathematical realm of bonuses will be useful to gamblers. If you'd like to win, all you need is to think and make calculations.

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