Mathematics of Bonuses
by Arthur Prudent
Online casino players know casino's offer various bonuses. Free
loads 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% payouts, a
bonus will allow you to make on average extra 2000 $ of stakes
($100/(10,95)=$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 payouts you will lose
on average $3000*(10,95)=$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 $1003000*0,005=$85 of the casino's
money.
The "sticky" or "phantom" bonuses:
More and more popularity in 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 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 bonus you have lost your $100 and that's
it, byebye. But with a bonus, even if it is a "sticky" one,
$100 are still on your account, which can help you worm out of
the situation. A possibility to win back the 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" bonuses one needs to use the strategy
"playanallornothing game". Really, if you play little
stakes, you will slowly and surely lose because of the negative
math expectancy in games, and the bonus will only prolong agony,
and won't help you 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. 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 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*(500150)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:
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 (1025%) 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 3,5fold
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 abovementioned
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 game we
need to have thousands of dollars on our account, but we
certainly can't blame casinos for dishonesty or gamblers for
being foolish.
Let's come back to our bonuses, to the most "freeload" ones
without any deposit. Of late one has been able to notice more
and more 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 account where you must win it back, like any bonus, usually
having run it 20 times in 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 bonus
back, you need to stake $10 000 in slots. We don't know the
rates of payouts in slots, they are not published by casinos
and make up on average about 95% (for various kinds they
fluctuate about 9098%). If we get at an average slot, then till
the end of the wager we'll have $50010 000*0,05=$0 on our
account , not a bad game... If we are lucky to choose a slot
with high payouts, we can await $50010 000*0,02=$300. Even
though the probability to choose a slot with high payouts is
50% (you have listened to the opinions of other gamblers since
by random choice this probability will make up hardly more than
1020%, for there are few generous slots), in this case the
value of a generous deposit free 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 bonus has decreased sevenfold.
I hope, this excursion into mathematics domain of bonuses will
be of use to gamblers if you want to win, you simply need to
think a little and make calculations.
