The Math of Gaming: What Exactly is “expected value”?

The theory of predicted or expected value is an integral theory to comprehend should you desire to construct a thorough comprehension of the maths and probability of gaming. Expected worth, in an unique circumstance of gaming, pertains to how a player can anticipate to do in certain casino over a series of bets. A fixed wager is, for instance, repeatedly wagering $20 on black in roulette.

The Computation of Expected Value
As a way to compute the predicted value of an unique gaming scenario, you should use the next formula:
[(odds of successful) x (sum won per wager) + (odds of losing) x (sum lost per wager)]

This can be tough to comprehend just in a brief summary, therefore let us use an illustration to help. Let’s say that the player is enjoying roulette and would love to sort out what the expected Value of a specific wager – let us say $10 on black – over an interval of time will be.

Thus, this means the odds of dropping is 18 in 38. The sum won per wager and therefore lost per wager is the same – that’s, the sum of the wager, which is $10. So, let us swap these amounts into the equation above:

This amount of -0.526 symbolizes the reality that if a participant makes a wager of $10 repetitively, they can theoretically anticipate to drop $.053 each time he makes a $10 wager. This reduction is incurred due to the house advantage that casinos run with, which assures they make a revenue in the long term. If you need more details about the home border, assess our Frequently-Asked Question What’s the-house Advantage?

If you desire to discover the predicted value for a string of indistinguishable, then additionally you will have to have some familiarity with probability concept, since odds are a vital part of the equation which is used to determine expected value. To get a detailed comprehension of likelihood theory, you’ll be able to assess our Frequently-Asked Question, What’s likelihood theory? Or do some investigation of your own. We can give you a simple description here. Probabilites can be expressed as likelihood – for instance, one in 10. In this instance, we might split 18 by 38 and locate the odds of touchdown on black in any specified twist of a roulette wheel is about 0.474 or about 47%

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Math vs. Opportunity in Gaming

All games of possibility take care of odds that one occasion will happen versus several other occasions. This likelihood is called the likelihood. What this means is that one-out of every two-times, in a lengthy string of trials, the coin will reveal heads. Likewise, the odds of tails is the same. The likelihoods are thus said to be even.

We shall see that it’s six faces, each bearing a distinct amount from one to 6, if we analyze a die which is used in the sport of craps. Subsequently the odds of any one amount coming up is one-out of six trials, if the die is an ideal cube, totally balanced.

If you’re to perform a gambling casino, which performed this sport of throwing one die, and you were to spend off the victor at 5-to-1, you’dn’t drop any cash, but would go broke from spending the operational expenses. What retains the casinos open is they constantly pay off at some thing less compared to the authentic likelihood. They continue roll up this “house percent” on every wager, which is how they make their cash.

Some illustrations follow:

In roulette, the likelihood against any unique amount coming up are 37-1, because there are thirty eight amounts on the American wheel (0,00, and 1-36). The casino pays this wager at 35-1. Thus, theoretically, and in the end, every time a wager is produced on an amount, the home is gathering 2/38ths, or 5.26%!

In craps, the chances against rolling a seven are 30:6; that’s, there are thirty six potential mixtures of dice, six which identical a total of seven. Seven arises, and if without a doubt on a seven being thrown, the household is only going to pay four-times to you your wager instead of five occasions. In this situation the home percent is 1/6 or 16.67%.

Where the quantity of the different symbols of each type on the device limits the odds of the various types of returns the same scenario exists with slots.

You realize after all of the stakes are in the chances are totaled, if you wager at the race track. The monitor splits the remainder, by totalizator device, among the victor and takes its percent off the best.

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Poisson Distribution – Introduction

A Poisson distribution is the chance of the amount of events that occur in a specified period. The events occur independently of one another and once the expected number of events is known. The symbol denotes the predicted number of events that happen during the period. The chance that you will find just x occurrences in the interval can be determined by the formula –

– where e is a constant equal to about 2.71828 (the base of the natural logarithm system), is the anticipated number of events that occur in the interval, x is the real number of events that occur in the interval, and x! is the factorial of x.

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What’s the Difference Between Independent Events and Dependent Ones

On our site we explain about Probability Theory by illustrating that the likelihood of an event occurring (say, rolling a four on a dice), is equivalent to the number of ways that that event can occur (in this case, 1), divided by the total number of those events which are possible in the given scenario (in this case, six). Therefore, the chance of rolling a four is 1 divided by six, or 0.1667. The different methods for working out the likelihood of an event depend greatly on whether or not the events are independent or dependent — but what is intended by this?

Defining Dependent and Independant Events

An independent event is one whose result is perhaps not predicated on the outcome of another event, and one whose outcome does not influence the outcome of another event, whereas a dependent event is the opposite of an independent event in that its outcome does affect, or is influenced by, the outcome of another event.

Being an instance of independent events: if we have two dice (see the mathematics of craps for more information on dice mathematics) and we roll them both, the value that is rolled on the first dice is not affected by the value that is rolled on the second dice, and the first dice’s value also does not play a part in determining the value on the second dice. Yet another instance of independent events is two throws of exactly the same coin — the likelihood of the next throw landing on heads or tails is constantly 0.5 (in other words, 1 in 2), no matter whether the very first throw landed on heads or tails. Dependant events are largely involved by the mathematics of poker, because the quantity of cards from which to draw every time to decreases a card is dealt, meaning that the chance of drawing a particular card from the deck increases for each time that that card is not dealt.

The probability of two separate events happening at exactly the same time (dice) or consecutively (two throws of just one coin) may be the probability of these events multiplied with one another. Therefore with the dice, where in fact the probability of throwing any value is 0.1667, we are able to work out that the probability of, as an example, throwing a 2 on the main one dice and a 1 on the other is 0.1667 x 0.1667 = 0.027. With the coins, the probability of throwing a heads and then the tails could be 0.5 x 0.5 = 0.25 (i.e. a one in four chance).

A good example of a scenario where independent events use in betting may be the lottery. People usually believe that if they stay with a particular group of numbers for long enough, the probabilities of these numbers coming up improve with every time that they do not appear. This is simply wrong, as each lottery draw is definitely a completely independent event — that is, previous lottery draws don’t have any influence on subsequent lottery draws. Hence, if you are thinking about winning the lottery, your opportunities are the same no matter whether new numbers are chosen by you every week or if you stay with the same group of numbers!


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The Gambler’s Fallacy

Imagine you toss a coin 10 times, what results would you expect?  5 Heads and 5 tails perhaps?  What if you actually tossed 2 heads and then the next 8 flips all turned on to tails?  Many people would now (incorrectly) expect the next flip to be heads in order to even out the distribution.  After all nobody would expect 8 tails in a row to turn into 9 tails in a row.

The Gamblers Fallacy

This idea that the next event is controlled by what’s come before is commonly known as the Gamblers fallacy. The assumption is that any short term event will behave in a way that matches the long term expectations.

If we tossed our coin thousands and thousands of times, we would expect the result to end up around 50% heads and 50% tails. The idea that the coin is more likely to end up as tails after a long line of heads is mistaken.

The single important fact with this, is that each event is independent. Each coin toss is in no way affected by the previous one, there is no obligation or effect on the result from previous results. Never think an event is due, never place any money on a bet that suggests that an event is due either. See what happens in this video about online roulette , it helps people appreciate that long streaks are possible simply because each event is independent.

When you toss a coin unless you’re cheating the result is completely random.  It is also why unless you can directly effect or control the mechanics of this event then any system is liable to fail, online casino players have been trying to figure out a way just like the physical players.

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Mathematics Supporting the Game

The application of the theory of probability to any circumstance such as picking a lottery ticket or playing a game of chance is actually quite a simple process.  The reason is that usually a finite sample space can be attached to any such game.

The finite sample space and indeed the randomness of any event which happens in that game (e.g drawing a card, spinning a wheel etc) allows us to put together a simple probability model.   This model in turn allows us to find the real numerical probabilities of any event which is associated with that particular game.

We will see that using the classical definition of probability that we can reduce even the most complex event into its simpler elementary events.

One of the games we will use the most in this site is the popular game of chance – roulette.  The main reason is that it is probably the easiest game with respect to probability calculations.  The main advantage it has is that all the elementary events are all pretty one-dimensional – these are the numbers on the roulette wheel.

The only other game that is comparatively as simple is dice, but these usually have many more combinations than a single spin of the wheel.  We will though occasionally wander into the world of other games.  Such as Bridge (my favorite card game), poker and blackjack.  The probabilities do become a little harder to calculate but we will try and use simple examples.

We hope this site will develop to become a useful resource for people learning about mathematics particularly applying probability theory to things that happen in real life!

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