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Online Casinos. Mathematics of Bonuses.

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Online casino players know that the latter ones offer various bonuses.
"Free-load" looks attractive, however, are they really useful these

casino bonus, cash back casino bonus, casino sticky bonus

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Online casino players know that the latter ones offer various casino
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 casino 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 casino 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
casino 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 casino bonus will help you, it can really be considered
free money. If you play casino 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
casino games, forbidden by casinos' rules for winning back bonuses. In
the majority of online 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 online 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 casino 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.

The "sticky" or "phantom" bonuses:
More and more popularity in online casinos is gained by "sticky" or
"phantom" bonuses - the equivalent of lucky chips in real casinos. The
amount of bonus is impossible to withdraw, it must remain on the account
(as if it "has stuck" to it), until it is completely lost, or annulled on
the first withdrawal of cash means (disappears like a phantom). At first
sight it may seem that there is little sense in such a casino bonus - you
won't get money anyway, but it's not completely true. If you win, then
there is really no point in the bonus, but if you have lost, it may be of
use to you. Without a casino bonus you have lost your $100 and that's it,
bye-bye. But with a bonus, even if it is a "sticky" one, $100 are still
on your casino account, which can help you worm out of the situation. A
possibility to win back the casino bonus in this case is a bit less than
50% (for that you only need to stake the entire amount on the chances in
roulette). In order to maximize profits from "sticky" casino bonuses a
casino player needs to use the strategy "play-an-all-or-nothing game".
Really, if you play little stakes, you will slowly and surely lose
because of the negative math expectancy in casino games, and the bonus
will only prolong agony, and won't help you win. Clever casino players
usually try to realize their casino 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 casino players use progressive strategies
of Martingale type. It is recommended to fix the desired amount of your
gain, for example $200, and try 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 casino 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 casino 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 casino games the results will be

The cash back bonus:
There is a seldom encountered variant of a bonus, namely return of
loosing. 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 loosing
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.

I will allow myself a short remark, slightly digressing from the main
subject. On a casino forum one of the gamblers started to claim that
tournaments were not fair, arguing it in the following way: "No normal
person will ever make a single stake within the last 10 minutes of the
tournament, which 3,5-fold surpasses the prize amount ($100), in
nomination of a maximal losing, so as to win. What is the point?"

And really does it make sense? The situation is very similar to the
variant with return of losing. If a stake has won - we are already in the
black. If it has lost - we'll get 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. Yes, we may lose $250 today, but shall
win $350 tomorrow, and over a year playing every day, we'll accumulate
pretty 365*$44=$16 000. Having solved a simple equation, we'll find out
that stakes up to $1900 are profitable for us! Of course, for such a
casino game we need to have thousands of dollars on our account, but we
certainly can't blame casinos for dishonesty or gamblers for being

Let's come back to our casino bonuses, to the most "free-load" ones-
without any deposit. Of late one has been able to notice more and more
casino advertisements promising up to $500 absolutely free of charge,
without any deposit. The pattern is the following - you really get $500
on a special account and limited time for play (usually an hour). After
an hour you get only the amount of your gain, but still not more than
$500. The gain is transferred on a real casino account where you must win
it back, like any casino bonus, usually having run it 20 times in casino
slots. $500 free - it sounds attractive, but what is the real price of
the bonus? Well, the first part - you need to win $500. Using a
simplified formula, we can see that probability of winning is 50% (in
practice, it is certainly even smaller). The second part - we win the
casino bonus back, you need to stake $10 000 in casino slots. We don't
know the rates of pay-outs in casino slots, they are not published by
online casinos and make up on average about 95% (for various kinds they
fluctuate about 90-98%). If we get at an average slot, then till the end
of the wager we'll have $500-10 000*0,05=$0 on our casino account, not a
bad game... If we are lucky to choose a casino slot with high pay-outs,
we can await $500-10 000*0,02=$300. Even though the probability to choose
a slot with high pay-outs is 50% (you have listened to the opinions of
other gamblers since by random choice this probability will make up
hardly more than 10-20%, for there are few generous casino slots), in
this case the value of a generous deposit free casino bonus amounts to
$300*0,5*0,5=$75. Much less than $500, but still not too bad, though we
can see that even with the most optimal suppositions the final amount of
the casino bonus has decreased seven-fold.
I hope, this excursion into mathematics domain of online casino bonuses
will be of use to gamblers - if you want to win, you simply need to think
a little and make calculations.

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