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Online casino players are aware that the latter ones provide a variety of bonuses. "Free-load" appears appealing, but are they really useful these bonuses? Are they profitable for gamblers? This is a question that depends on many factors. Mathematical knowledge can assist us in answering this question.Let's begin with the typical bonus on deposit. You deposit $100 and get another $100. This is feasible after you stake 3000. It is an example of a bonus that you can get on the first deposit. While the amount of a bonus or deposit can vary as well as the stake rates. But there is one thing that is for certain: the bonus amount is still able to be withdrawn following the wagering requirement. In general, it is impossible to withdraw any funds.If you intend to play in the online casino for a long duration and with a lot of intensity you are a player, this bonus could aid you, and it could 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)=$2000), after that the amount of bonus will be over. There are a few pitfalls, for example, if you want to simply take an experience at a casino without having to play for long and you are a fan of roulette or other gamesthat are forbidden by casinos' rules to win back bonuses. If you do not wager in any of the permissible games, the majority of casinos will not let you withdraw cash. Bonuses can be won when playing roulette or blackjack however, only if meet the minimum stakes of 3000. In the 95% of all payouts that you'll lose an average of 3000$ (1-0,95) = $150. In other words, you not only lose the bonus, but will also be able to take from your wallet $50. In this case it is better to not accept the bonus. If blackjack or poker can win back the bonus by earning a profit of 0.5 percent, it's possible to expect that you'll receive $100-3000*0,005=$85 after you have won back the bonus."Sticky" as well as "phantom" bonusesCasinos are becoming more popular because of "sticky" and "phantom bonuses. These bonuses are equivalent of lucky chips in a real casinos. The amount of the bonus is impossible to withdraw and must stay on the account (as when it "has stuck" to it) until it's completely lost, or annulled upon the first withdrawal cash means (disappears as if it were an illusion). On first glance, it might seem that there is little value in bonuses - you don't be able to withdraw money at all however, this isn't correct. The bonus won't be worth it if you win. But, if you lose, it may be beneficial. Already, you've lost $100, without a bonus. But with a bonus, even if it's one that is "sticky" one, you will find that $100 remain on your account, which can aid you in escaping the circumstance. A possibility to win back the bonus is around 50 percent (for it is only necessary 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". You'll lose slowly and surely if you stake small amounts. The math expectancy that is negative of games means that you will not get any bonuses. http://www.demoscenebook.com/books/ 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 amount you want to winnings, such as $200, and then try to win it by 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:It is not often seen type of bonus, namely return of losing. There can be singled out two options - either the complete refund of the deposit lost, at this the returned money usually is to be returned as an ordinary bonus, or a partial return (10-25 percent) of the loss over the fixed period (a week or month). http://www.demoscenebook.com/history-of-craps-and-keno/ is similar to a "sticky bonus" which is useless in the event of winning, but helps if you lose. Calculations in math will also be identical to "sticky" bonus, and the strategy is similar - we risk trying to win as much as we can. If we do not win and we have lost then we are able to play again with the help of that money back, thus taking the risk to a minimum. Casinos with games offer a partial return on losing to gamblers who have a high level of activity. If you gamble on blackjack using math expectancy - 0,5%, then after you have staked $10,000, you'll lose an average of $50. If you earn 20% of the money, the amount of $10 is returned to you, that is your loss will be $40, which is comparable to the growth in math expectancy up to 0,4 percent (ME with return=theoretical ME the game * (1-% of return). But, from the bonus, you can also gain benefit, for that you will need to play less. We make only one but a high stake, for example $100, with the same stakes in roulette. The majority of cases we also win $100 and 51% of the time we lose $100, however at the end of the month we win back 20%, which is 20 dollars. 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 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 every month.I'd like to provide a short remark. I'm a little off-topic. In a forum for casinos, one of the gamblers started to assert that tournaments were not fair, arguing it in the following way: "No normal person will ever stake a single penny within the last 10 minutes of a tournament, which 3,5-fold surpasses the prize ($100), in nomination of a maximal losing, so as to win. What's the point?<img width="457" src="">And really does it make sense? It's identical to the scenario with return of losing. If a stake is successful it is already in the black. If it is lost, we'll win 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. Sure, we could lose $250 today, but we will win $350 next day. Over a year of playing each day, our total earnings will be pretty impressive at 365*$44=$16 000. We'll see that stakes up to $1900 can be profitable for us after solving the simplest equation. Of course, to play such a game , we'll require many thousands of dollars in our accounts, but we certainly can't blame casinos for dishonesty or gamblers who are foolish.Let's look back at our bonus offers, especially the best "free-load" ones- without any deposit. One has noticed more and more ads promising $500 for free, without deposit. You can get $500 on an account that is unique, and only a certain amount of time to play (usually 1 hour). You will only get the amount you win after an hour, but no more than $500. The gain is transferred on an actual account, where you must win it back, like any bonus, usually having played it at least 20 times on slots. $500 for free sounds appealing however, what is the real price of the reward? First, let's look at the first step is that you must win $500. It is evident that the odds of winning $500 is 50% based on a simplified formula. However, in practice it is much lower. In order to win the bonus back You must bet at least $10 000 on slots. The pay-out percentages of slot machines aren't well-known. They range from 95 to 95%, but can vary between 90-98% for different types. The average slot gives us between $500 and 000*0.05=$0. This isn't an awful amount. If we happen to choose a slot with large payouts, we can await $500-10 000*0,02=$300. Although the chance to choose a slot with payouts that are high is 50 percent (you are probably familiar with the opinions of other gamblers , since by random choice this probability will make up hardly more than 10-20% since there are few generous slots) in this scenario, the worth of a large deposit-free bonus amount to $300*0.5%*0.5%=$75. Even though it's not $500, it is an impressive amount. However, we can observe that the bonus's total value has decreased sevenfold, even with the best possible estimates.I'm sure this trip into the maths of bonus will prove useful to gamblers - if you are looking to win, you just must think about it and make calculations.