Mathematics of Casino Games | Casino Guru Explains

We have all heard stories about players winning
millions in casinos using secret tricks and
not having to work another day, just enjoying
luxurious holidays drinking ice cold margaritas
on beautiful sunny beaches.
Well, those stories are fake.
The math behind casino games always favours
so there is no guaranteed way for players to win.
Even if you are lucky at first and won, if
you keep playing long enough, the math will
sooner or later turn against you and you will
lose your money.
Now, let me show you how the math of all casino
games works
and why the casino always profits in the end.
Casino games such as slot machines, roulette
or blackjack are called games of chance because
their outcome depends on luck.
You play them against casinos, not against
other players.
Regardless of your skill, the casino will
always have an advantage.
Let’s take a closer look at how it all works.
All casinos and casino games have certified
fair randomness, which means that the game
results really are random.
But remember; fair randomness doesn‘t make
a game fair.
Here is an example:
Imagine a game in which we flip a coin.
If it lands on eagle, I will give you one
However, if it lands on heads, you will give
me two dollars.
Even though the randomness is fair, the game
clearly isn’t.
Your chances of winning and losing are the
same, but the amount you can lose is twice
as high as the amount you can win.
So, does that mean you can’t win any money?
Well, you can, but only in the short term.
Even in our imaginary game, you can be very
lucky and win the first few flips, giving
you a couple of dollars in profit.
However, the incredibly one-sided odds of
the game will catch up with you very quickly.
Real games in actual casinos (whether land-based
or online) have better odds, which might not
cause you to lose money as fast as you would
in our made up game, but you can be sure that
you will eventually lose.
You just have to give it enough time.
Now that you know that all casino games are
unfair for players, let’s take a look at how
you can distinguish which of them are more
unfair and which are less unfair.
This is where the term ‘return to player’
(RTP) comes in.
It’s the long-term percentage of wagered money
that you get back from the casino in the form
of wins.
It essentially tells you how much of your
bets you can expect to get back from the casino
when playing a specific game.
Let’s go back to the game with a coin.
In this game, you have a 50% chance of winning
a dollar, and a 50% chance of losing two dollars.
This means you lose half a dollar on average.
Let’s reformulate the rules a bit to better
illustrate the RTP.
You bet two dollars and flip a coin.
If you win, you will get back your two dollars
and another one on top of that,
which is three dollars in total.
In other words, you pay two dollars for a
50% chance to win three dollars.
You end up with one-and-a-half dollars on
which is 75% out of the wagered two dollars.
This is the RTP of this game.
This means that you only get back 75% of each
bet in the long run.
It’s not that bad for most games though.
Most casino games have an RTP somewhere in
the range of 95% to 99%.
The remaining percentage forms the house edge:
the mathematical advantage casinos have over
players, which enables them to run a profitable
For example, roulette has a house edge of
Does this mean that you will lose 2.7% of
your money when playing roulette?
The answer to this question is yes and no.
It depends on how you play.
If you wager your money only once, you will
only lose 2.7% of it on average.
However, if you wager your money over and
over again, you will lose 2.7% of each bet
on average, which adds up quickly over time.
This is why we also need to differentiate
between the RTP of a game and the expected
return of a betting strategy.
The expected return of a betting strategy
describes how much money you can expect to
get back from a gambling session on average
when you follow a certain betting pattern.
It is never higher than the RTP of the game
you play, but it could be lower.
Much lower, actually, depending on how you
Let’s take a look at two examples, both played
European roulette with an RTP of 97.3%
and a 2.7% house edge.
In the first example, your strategy is to
place a 100 dollar roulette bet
and hope for the best.
If you win, you take your money and leave.
If you lose, you admit defeat and move on.
In this case, the expected return of your
betting strategy is the same as the game’s
RTP because you only wager your money once.
The expected statistical loss is therefore
$2.7, because the house edge is applied to
your money just once.
Imagine a second example.
Let’s say you walk into a casino
and start placing 10 dollar bets on the color red or black.
You play until you go through 100 game rounds
or until you lose everything.
In this case, you lose 22 dollars on average
because you wager much more money in total,
and lose 2.7% of each placed bet on average.
Okay, you already know how much money you
will lose, but how you lose it over time is
also important, as well as your chance of
hitting a lucky big win.
That’s where volatility, also known as variance,
comes in.
These two players play a game with the same
RTP, but with a different volatility.
The player on the left plays a low-volatility
His wins are frequent but low.
That’s why the size of his bankroll fluctuates
only slightly.
The player on the right plays a high-volatility
The majority of his game rounds end up in
a loss,
but he hits a big win from time to time.
Having said that, both players will eventually
lose everything if they keep playing.
So, does volatility even matter?
And if it does, which games are better?
High volatility games or those with low volatility?
It does matter.
It actually matters more than you might think.
Although the preferences of players might
high volatility is better from a mathematical perspective.
Here are a couple of examples to demonstrate
In the first one, you start with a bankroll
of 100 dollars and keep placing five dollar
bets on red or black until you play 100 game
rounds or lose everything.
Playing like this, you have a 35.6% chance
of ending up with more money than you started
with, but you will only win a small amount
in most of the ‘winning’ cases.
Your chance of ending up with more than 200
dollars is only 1.5%.
On the positive side, you will lose everything
before reaching the 100-spin mark in only
4.3% of cases.
On average, you will lose 13.2 dollars.
Now, let’s ramp up the volatility by keeping
the same bet size and placing bets on a specific
number instead of red or black.
When betting on color, you are quite likely
to double your bet, while bets on a number
give you a proportially smaller chance to
win 36-times your bet.
It is therefore pretty clear that the volatility
is higher.
In this example, your chance of ending up
in profit is lower.
It is 24.6% to be exact, which is 11% less
than before.
However, if you do win, you’ll win much more
on average.
Quite surprisingly, you are more likely to
end up with 500 to 1000 dollars
than 100 to 200 dollars.
All in all, you will lose 6.5 dollars on average.
On the negative side, your chance of reaching
the 100-spin mark is smaller than 25%.
This is a sign that we might have taken it
a bit too far in terms of volatility,
but that’s not an issue because higher volatility
allows you to place smaller bets and get the
same (or even higher) chance of winning big.
Let’s take a look at the same betting strategy,
but with a two dollar bet instead of a five dollar bet.
As you can see from the table, this strategy
is superior to the first one in almost all
aspects, with a chance to win money at over
46%, including a more than 13% chance
to end up with 200 to 500 dollars, and even a chance
of getting over 500 dollars.
The only thing that’s worse is the possibility
of reaching the 100th spin.
You are guaranteed to play at least 50 game
rounds, but need to win at least one of them
to have enough money for 36 more game rounds.
And if you win at least two spins, you will
reach the 100-spin mark.
Taking all of this into account, the possibility
of playing the full 100 spins is 61.4%.
In exchange for the higher risk of running
out of money prematurely, you get a much higher
possibility of winning a bigger amount, and
a smaller average loss of only 4.7 dollars
because the overall amount you wager is smaller
on average.
Before you ask…
To get all of these numbers, we ran a million-player
simulation for each betting strategy in order
to get reliable results.
What should you take away from this?
Higher volatility enables you to have the
same (or even higher) chance of winning big
even when placing smaller bets, whereas smaller
bets make you lose less on average because
the house edge is applied to less money overall.
Even though you lose less on average, you
also lose your money much faster in most cases.
That might sound bad, but it’s actually good
for your expected return because you are often
not given a chance to wager your money again
and again,
statistically losing a part of it each time.
To summarize, high volatility gives you a
better chance of getting a big win, but it
also increases your chance of losing your
entire bankroll, which is something you need
to keep in mind when deciding how and which
games to play.
Regardless of the level of volatility you
choose, remember that you are always playing
at a disadvantage.
You might be asking yourself, is there really
no way to beat the casino?
To be fair, there have been people who have
managed to beat casinos in the past by finding
a trick that worked.
And there are some strategies that you can
use to get the advantage on your side,
such as card counting in blackjack or bonus hunting
in online casinos.
However, these are very complicated to put
into practice and definitely aren’t for everyone.
Chances are, if you are good enough to use
these strategies, then you can make much more
money by investing your time into something
Everybody else will have to satisfy themself
with always playing at a disadvantage,
so keep that in mind the next time you enter a casino.
Because you are always going to lose money
if you keep playing, you might be asking yourself
whether you should be playing games of chance
at all.
The answer to this question depends on what
you want to achieve.
If your goal is to make money, you should
not be playing casino games because these
are not a way of making money or solving your
financial issues.
If you just want to have fun and see casino
games as a form of entertainment that you
are willing to pay for by playing at a disadvantage,
go for it, but remember to gamble responsibly
and only bet what you can afford to spend.
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