Notes

Online Casinos: The Mathematical Basis of Bonuses
Casino players who play online know that these casinos offer a range of bonuses. Although "Free-load" may seem attractive, they're not really worthwhile. Are they profitable for gamblers? Answering this question is contingent on a lot of conditions. The answer to this question is possible by math.

Let's begin with the typical bonus when you deposit. You transfer $100 and receive another $100. This is possible after you stake $3000. It is an example of a bonus that you can get on the first deposit. The size of the bonus and deposit can be different in addition to the required stake rates However, one thing remains in place - the amount of bonus can be withdrawn following the wager requirement. It generally is not possible to withdraw any funds.


The bonus is free money if you gamble online for a lengthy duration and keep playing. 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 some complications. For example when your objective is to simply have a peek at the casino, without spending a lot of time there, or you like roulette or other games which aren't permitted under bonus rules, you might be denied access to the bonus amount. If you aren't betting on any of the allowed games, most casinos won't allow you to withdraw money. If single player games on blackjack or roulette, and a bonus is won back only by playing slot machines, you must make the required stakes of $3000 and in the 95% payouts, you will lose on average $3000*(1-0,95)=$150. The loss is $50 and you also forfeit the bonus. In this scenario, it is better not to accept the bonus. If blackjack or poker can win back the bonus by earning a profits of 0.5 percent, it's likely that you will get $100-3000*0,005=$85 after you've earned back the bonus.
"sticky" or "phantom" bonus:

More and more popularity in casinos is due to "sticky" or "phantom" bonuses - the equivalent of luck chips in real casinos. It isn't possible to cash out the bonus. The bonus has to be kept on the account, like it "has stuck". It might appear as if bonuses are not worth the effort. It isn't possible to take any money out, but this isn't the case. The bonus is not worth the cost if you win. If you lose, it may prove useful. Without a bonus you have lost $100, and then you're gone. If the bonus was not "sticky", $100 will still be in your account. This could help you to wiggle out of the situation. The chance of winning back the amount of bonus is a bit less than 50 percent (for it is only necessary to stake the entire amount on the chances 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 little stakes, you will slowly and surely lose due to the negative math expectation in games, and the bonus will only prolong the pain, and will not help 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. Set the amount you want to earn, like $200, and then take the risk to make it. 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 variation of a bonus which is the return of lost. It is possible to distinguish two variations - the full refund of the deposit lost and the amount is usually paid back as with any other bonus or a portion (10-25 percentage) of the losing during the specified time (a week, a month). In the second case, the situation is practically identical to that of the "sticky" bonus. In the event that we win, there's no need for the bonus, however it helps in case of loss. The "sticky bonus" mathematical calculation will be analogous. The strategy of the game is identical that we bet to win as frequently 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 decreasing the risk. Casinos with games offer some kind of compensation for losses to gamblers who have a high level of activity. You'll lose about $50 if you play blackjack with an average math expectation of 0.5 percent. You will receive $10 back if you lose 20 dollars. This is equal to an increase in math expectancy of 0.4 percent. However, from the given bonus you will also get from the fact that you'll need to play less. On the same stakes as in roulette, we play one, but it's an enormous stake. We can win $100 in 49% of cases and $100 is won by 51%. However, we lose $100 in 51% of the cases. At the end of each month, we earn back 20 percent of the $20 we won. As a result the effect is $100*0,49-($100-$20)*0,51=$8,2. The stake has a positive mathematical probability. But, the dispersion is high and we'll only be able to play this way once each week or every month.

I'll allow myself a short remark, slight deviation from the main topic. One forum member claimed that tournaments weren't fair. He said, "No normal person will ever put a stake in in the final 10 minutes." This 3,5-fold exceeds the prize amount ($100) in the nomination of maximum losing, so that you won't lose. What is the point?"

It makes sense. The situation is quite similar to the one with return of losing. We're in the black if the stake is won. 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 be losing $250 right now, however we we'll win $350 tomorrow, and over a year playing each day, we'll earn 16 000 dollars. It's clear that stakes as high as $1900 can be profitable for us if we solve an easy equation. We'll need thousands on our accounts for this kind of game, but we can't blame the casinos for being shady or naive.

Let's look back at our bonus offers, especially the highest "free-load" ones- with no requirement for any deposit. Recently, one has noticed an increasing number of ads promising up to $500 absolutely for free, and without any deposit. You can get $500 on an account that is unique, and you have a time limit to play (usually an hour). The only thing you will get is the amount you win after an hour, but not over $500. The cash is transferred to an actual account, where you must get it back as any other bonus, typically after having played it at least 20 times in slots. This sounds fantastic, but what's the actual value of this bonus? The first thing to consider is is that you must win $500. Based on a simplified formula, we will see that the probability of winning is 50 percent (in the real world, it's certainly even smaller). To win the bonus back You must be able to stake at least $10 000 on slots. The pay-out percentages of slot machines aren't well-known. They average around 95% and fluctuate between 90-98 percent for various types. A typical slot can give us $500-10 000*0.05=$0. This isn't an awful amount. If we're lucky enough to pick a slot with payouts that are high, we could look forward to $500-10 000*0,02=$300. The chance of selecting one with the highest payout is 50%. But, you've read the comments of other gamblers that this probability will not exceed 10-20%. In this case the bonus for depositing is generous of $300*0.5*0.5=$75. Even though it's not $500, it's still a good amount. However, we can observe that the bonus's total value has dropped sevenfold even with the most accurate estimates.

I am hoping that this investigation into the maths of bonuses can prove beneficial for gamblers. If you want to be successful, all you need is to think and make calculations.

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