# Online Casinos: Mathematics of Bonuses

Online casino players know that the latter ones offer various bonuses. “Free-load” looks attractive, however, are they really useful these bonuses? Are they profitable for gamblers? The answer to this question depends on a lot of conditions. Mathematics will help us answer this question.

Let’s begin with an ordinary bonus on deposit: you transfer \$100 and obtain \$100 more, which it will be possible to get having staked \$3000. It is a typical example of bonus on the first deposit. The sizes of a deposit and bonus can be different, as well as the required stake rates, but one thing remains unchangeable – the amount of the bonus is accessible for withdrawal after the required wager. Till this moment it is impossible to withdraw money, as a rule.

If you are going to play in the online casino for a long time and rather insistently, this bonus will 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, for example, if you simply want to have a look at a casino, without playing for a long time, if you prefer roulette or other games, forbidden by casinos’ rules for winning back bonuses. In the majority of casinos you won’t be allowed to withdraw money or will simply return a deposit, if a wager is not made on the games allowed in the casino. If you are keen on roulette or blackjack, and a bonus can be won back only by playing slots, make the required \$3000 of stakes, in the course of 95% of pay-outs you will lose on average \$3000*(1-0,95)=\$150. As you see, you not only lose the bonus but also take out of your pocket \$50, in this case it is better to refuse the bonus. Anyway, if blackjack and poker are allowed for winning back the bonus with a casino’s profit only about 0,5%, so it can be expected that after winning back the bonus you will have \$100-3000*0,005=\$85 of the casino’s money.
“sticky” or “phantom” bonuses:

The cash back bonus:

There is a seldom encountered variant of a bonus, namely return of losing. There can be singled out two variants – the complete return of the lost deposit, at this the returned money usually is to be won back like with an ordinary bonus, or a partial return (10-25%) of the losing over the fixed period (a week, a month). In the first case the situation is practically identical to the case with a “sticky” bonus – if we win, there is no point in the bonus, but it helps in case of losing. Math calculations will be also analogous to the “sticky” bonus and the strategy of the game is similar – we risk, try to win as much as possible. If we are not lucky and we have lost, we can play with the help of the returned money, already minimizing the risk. Partial return of the losing for an active gambler can be regarded as an insignificant advantage of casinos in games. If you play blackjack with math expectancy – 0,5%, then, having made stakes on \$10 000, you will lose on average \$50. With 20% of return \$10 will be given back to you, that is you losing will amount to \$40, which is equivalent to the increase in math expectancy up to 0,4% (ME with return=theoretical ME of the game * (1-% of return). However, from the given bonus can also be derived benefit, for that you need to play less. We make only one but a high stake, for example \$100, on the same stakes in roulette. In 49% of cases again we win \$100, and 51% – we lose \$100, but at the end of the month we get back our 20% that is \$20. As a result the effect is \$100*0,49-(\$100-\$20)*0,51=\$8,2. As you see, the stake then has positive math expectancy, but dispersion is big for we’ll be able to play this way rather seldom – once a week or even once a month.