What Are the Odds of Winning on a Slot Machine?

Saturday, 02nd September 2017

Slot machine odds used to be easy to calculate. When you’re dealing with three reels, ten symbols on each reel, and a limited pay table, then it’s just a simple math problem. But the rise of electromechanical slot machines and (later) video slots added some complexity to the situation.

How Probability Works

Probability has two meanings. One is the likelihood of whether or not something will happen. The other is the branch of mathematics that calculates that likelihood. To understand the odds as they relate to slot machines (or any other gambling game), you have to understand the basic math behind probability.

Don’t worry though. The math isn’t hard. Probability involves addition, subtraction, multiplication, and division, all of which you learned in middle school.

The first principle of probability is that every event has a probability of between 0 and 1. If something has no chance of ever happening, then its probability is 0. If something will always happen, no matter what, then its probability is 1.

Probability is, therefore, always a fraction. It can be expressed in multiple ways, as a decimal, as a fraction, as a percentage, and as odds.

A simple example is a coin flip. The probability of getting heads when you flip a coin is 50%. That’s common sense, but how is it determined mathematically?

You simply take the total number of possible outcomes, and divide the outcome you’re trying to determine the probability of it by that number. There are two possibilities when flipping a coin, heads or tails, but only one of them is heads. That’s 1 divided by 2, which can be expressed as ½, 50%, 0.5, or 1 to 1 odds.

Odds are expressed as the number of ways something won’t happen versus the number of ways that something will happen. For example, if you’re rolling a single six-sided die, and you want to know the odds of rolling a six, you’re looking at 5 to 1 odds. There are five ways to roll something other than a six, and only one way of rolling a six.

When you want to determine the probability of multiple things happening, you use addition or multiplication, depending on whether you want to determine whether one OR the other event will occur, or whether you want to determine whether one event AND the other event will occur.

If you’re looking at an “OR” question, you add the probabilities together. If you’re looking at an “AND” question, you multiply the probabilities by each other.

So if you want to know what the probability of rolling two dice and having one or the other come up with a six, you add the probabilities together. 1/6 + 1/6 = 2/6, which is rounded down to 1/3.

If you want to know the probability of rolling two dice and having BOTH of them come up six, you multiply the probabilities. 1/6 X 1/6 = 1/36.


How Slot Machine Odds USED to Work

Early slot machines were mechanical devices. They had three metal reels that had ten possible stops each.

To calculate the odds of a single symbol appearing on a reel, you just divide the one symbol by the total number of potential outcomes. So if you had one cherry on a reel, your odds of hitting that cherry were 1/10, or 10%.

To calculate the odds of getting three cherries, you multiple 1/10 X 1/10 X 1/10 and get 1/1000, or 0.1%.

If the odds of hitting that symbol are the same as all the others, then you have 10 possible jackpots you can win, which means that your chances of winning SOMETHING are 10/1000, which is 1%.

Most people wouldn’t play a slot machine that lost 99 times out of 100, though, so slot machine designers added additional, smaller prizes for getting two symbols out of three for certain symbols. And as long as they paid out less in prizes than the odds of hitting those jackpots, then those slots are guaranteed to make a profit in the long run.

For example, if a prize for hitting three cherries was $1000, you’d be playing a break-even game, but if the prize were $750, it’s easy to see how the casino would be guaranteed a profit. The difference between the odds of winning and the payout odds is where the casino makes its money.


How Slot Machines Work Now

Modern slot machines use a computer program called a random number generator to determine the outcomes of the various spins of the reels. This creates an imaginary reel with a number of symbols limited only by the program in question.

A mechanical slot machine with 256 symbols per reel would be huge, too large to play, much less to build. But a computer can create an imaginary reel with 256 symbols per reel and take up no more space than an iPod Shuffle.

To make things even more interesting and entertaining, slot machine designers can program different probabilities for each symbol to come up. Most symbols might come up once every 256 spins, but others might come up twice as often, while still others might only come up half as often.

This enables slot machine designers and casinos to offer slot machine games with far larger jackpots than they were able to when they were limited by mechanical reels. And they’re able to offer these large jackpots and still generate a healthy profit.


How Does This Relate to Payback Percentages?

The payback percentage is the amount of money that the slot machine is designed to pay out over an enormous number of spins. This number is almost always less than 100%. The difference between 100% and the payback percentage is the house edge, and that’s where the casino makes its profits.

A simple example can help illustrate how this works. Suppose you have a slot machine with three reels with ten symbols on each, and it only pays out when three cherries hit. The odds of winning that jackpot, as we determined earlier, is 1/1000.

If we set the jackpot as $900, and charge $1 per bet, the payout percentage for that game will be 90%, or $900/$1000. Of course, no one would play a slots game which only paid out once in every 1000 spins, which is why there are various smaller payouts programmed in.

There’s no way to tell what the payback percentage on a particular game is unless you have access to the par sheet for that machine. Casino management has that information, but players never have access to that info.

The best slot machine odds are almost always found in real casinos. If you see slot machines in an airport or a bar, be aware that the payback percentages on those games is much lower than you’ll see in a real casino.


Odds for Dice Outcomes in Craps

Tuesday, 22nd August 2017

When someone discusses craps odds, they’re discussing one of two things—the odds of rolling a certain number, or the payout for a particular bet. Odds are one way of describing a probability, but they’re also a way of describing how much a bet pays.

This page explains both types of craps odds. We refer to payout odds as the number that a bet pays off, and true odds as the probability that a given outcome will appear. The difference between the true odds and the payout odds is the house edge, which is the number that explains how the casinos stay so probable.


The Basics of Probability

Probability is that branch of mathematics that deals with the likelihood of something happening (or not happening). An event’s probability is always a number between 0 and 1, but that number can be expressed in multiple ways.

A simple example of a probability is a coin flip. The probability of the result being heads is 0.5, because half the time, that’s what will happen. 0.5 can be expressed also as 50%, ½, or 1 to 1.

That last expression of the probability is the one we’re most concerned with on this page, because that’s an expression of the odds.

The equation for calculating a probability is to divide the number of ways something can happen by the number of total ways it could happen. When rolling dice, you can calculate the odds of rolling a 1 by dividing 1 by 6. There are 6 possible outcomes, but the one you want to know is the chance of rolling a 1.

1/6 can also be expressed as 0.167 or 16.7% or 5 to 1. When expressing a probability as odds, you compare the number of ways something won’t happen with the number of ways something can happen. There’s only 1 way to roll a 1 on a single die, but there are 5 ways to roll something else, so the odds are 5 to 1.

When you’re calculating multiple probabilities, you add the probabilities together when you want to know the odds of event A OR event B happening. You multiple them by each other when you want to know the odds of event A AND event B happening.

For example, if you want to calculate the probability of rolling a total of 2 on 2 dice, you would multiply the probabilities of rolling a 1 on the first die by the probability of rolling a 1 on the second die. 1/6 X 1/6 = 1/36, which can be represented as odds of 35 to 1. (You’re calculating the odds of rolling a 1 on die A AND the odds of rolling a 1 on die B.)

On the other hand, if you wanted to know the probability of rolling a 1 on either of the two dice, you’d ADD the two probabilities together, and you’d get a result of 1/6 + 1/6, or 2/6, which can be reduce to 1/3. That would be expressed in odds as 2 to 1.


True Odds

When you discuss the odds of something happening, you’re discussing the true odds, or the probability, that something will happen. The difference between the true odds and the payout odds is what creates an edge for the casino. Casinos wouldn’t make a profit if they paid bets off at their true odds; they’d only break even. And like any other business, casinos exist to make a profit.


Payout Odds

So every bet in a casino pays out at less than true odds, except for one, which we’ll discuss later. For example, if you make a bet on something that has a 3 to 1 chance of happening, and the casino pays out at 2 to 1 on that bet, the casino will make a profit in the long run.

Suppose in a mathematically perfect simulation that you place four bets of $1 each on something that has a probability of occurring of 3 to 1. You would win once and lose three times. If you lose $1 on your three losses, and you win $2 on your single win, how much money did you net? You lost $1.

For every dollar that you wagered in that scenario, you lost an average of 25 cents.

That’s the house edge in a nutshell.


The House Edge

The house edge is usually expressed as a percentage of each bet that you can expect to lose over the long run. In the example above, the house edge was 25%, which is huge.

The house edge on most casino games is between 1% and 10%, but in craps, you’ll find some of the best bets and some of the worst bets in the casino.


The Best Odds in Craps

The best bets at the craps table are the ones with the lowest house edge, and luckily, those are also the simplest bets you can make. These bets include the pass bet, the don’t pass bet, the come bet, the don’t come bet, taking odds, and laying odds.

The house edge on the pass bet and the come bet is 1.41%, which means that for every $100 you wager, you should expect to lose, in the long run, an average of $1.41.

The house edge on the don’t pass and the don’t come bet is 1.36%, which means that for every $100 you wager, you should expect to lose, in the long run, an average of $1.36.

When you take odds or lay odds, your bet pays out at true odds. This means the house edge is 0, making this the best bet in the casino. The only catch is that in order to take or lay odds, you have to make a pass or don’t pass bet first.


The Worst Odds in Craps

The worst bets at the craps table are the complicated bets. They have the highest house edge, and when we say the house edge is high, we mean that it’s staggering.

The craps table features countless proposition bets of varying complexity, but here are a few examples of bets with bad odds in craps.

The Big 6 and Big 8 bets offer a house edge of 9.09%. That’s absurd when you consider that you can place the same bet as a “place bet” and only face a house edge of, at most, 6.67%.

Hardway bets also offer lousy odds. The house edge is either 9.09% or 11.11%, depending on which hard total you’re wagering on.

The Any Seven bet is another doozy. The house edge on this one is a whopping 16.67%.

Any time you find a game with bets with a house edge that ranges between 1.36% and 16.67%, you should educate yourself about which bets are offer the best odds and which ones offer the worst odds.


What are the Odds in Blackjack and How Can I Improve Them?

Monday, 21st August 2017

Blackjack offers some of the best odds in the casino. Everyone knows that already. But what does it mean to say that one casino games offers better odds than another casino game? This page goes into some detail about how to measure the odds of a casino game. Then it continues with an examination of the specifics related to the game of blackjack.


Odds and Probability Explained

Probability is the branch of math that covers how likely it is to achieve a certain result. Any possible result can be measured as a number between 0 and 1, with a 0 meaning that it will never happen and a 1 meaning that it will always happen. It’s probably easier to think of this number as a percentage, so something with a 0% chance of happening will never happen, while something that will always happen has a probability of 100%.

If you add the probability of something happening with the probability of something not happening, you will always get a total of 100%. So it’s always possible to calculate that number backwards. For example, if you know that there is an 80% chance of something not happening, you also know that there is a 20% chance of it happening, and vice versa.

Probabilities are sometimes explained as odds. This is just a way of expressing a probability as the number of possibilities of it failing versus the number of possibilities of it succeeding. For example, if you want to express the odds of getting an ace with the next card you draw out of a fresh deck of playing cards, then you know that the odds are 12 to 1. (For every ace in the deck, there are 12 cards that are NOT aces.)

Casinos make their money by paying out wagers at odds that are less than their odds of happening. For example, on a single number bet in roulette, the odds of winning are 37 to 1. (There are 38 different numbers, and you only bet on 1 of them in this example.) But that bet only pays out at 35 to 1.

In a mathematically perfect set of spins, you’d win once and lose 37 times. If you’re betting $1 per spin, then you’ve lost $37, but you’ve won $35 the one time you got a winning spin. The casino gets the other $2, and that’s how they stay in business.

All casino games, including blackjack, work on this seemingly-simple principle. In fact, it’s relatively easy to determine a percentage of each bet that you make that the casino can expect to receive over time. This percentage is called the house edge.

In the roulette example above, the house edge is 5.26%. That’s the amount of each bet that will be lost over an infinite number of trials. Of course, in the short term, anything can (and often will) happen, but as you get closer to an infinite number of trials, the closer the results become to the mathematical expectation.

The house edge can be as high as 25% on slot machines in Nevada, or as low as 0.5% on blackjack in the same state. But achieving that 0.5% number requires a certain amount of skill and strategy. Players who just use “common sense” or who “play their hunches” face a house edge of closer to 2-4%.


Basic Strategy Is the Key to the Best Blackjack Odds

The key to maintaining the lowest possible house edge on blackjack is to use basic strategy every time you play. By doing so, you maximize your expectation and minimize the casino’s expectation. Basic strategy is the mathematically perfect play in every possible blackjack situation.

This might seem like an unearthly number of scenarios to memorize, but it’s actually easier than you think. The dealer’s upcard is limited to 10 possibilities. Your hand is limited to 20 different totals. Since many hands will be played with the same strategy as many others, memorizing basic strategy is far easier than you’d think.

Many websites and almost all blackjack books offer basic strategy charts which can help visual learners memorize the correct strategy. The conditions and house rules can affect a handful of strategy decisions, but even then, it’s a tiny percentage of the potential situations you might face. Learning one basic strategy is better than learning no basic strategy at all. In other words, don’t get hung up on making the occasional error because of local playing conditions.


Playing Conditions and Rules Variations

The other key to getting the best odds in blackjack is to make intelligent choices about which blackjack games you’ll play. The biggest variation that you need to watch out for is the payout on a natural 21. At most casinos, a natural 21 pays out at 3 to 2 odds, but you’ll often find games that pay out at 6 to 5 odds instead.

The difference in that single rules variation is tremendous. In fact, it gives the casino an additional edge of 1.39%. In most blackjack games, this doubles or even triples the house’s edge. The easiest way to avoid that additional “tax” on your game is to just say no to any blackjack game that doesn’t pay out at least 3 to 2 for a natural.

Other variations in the rules can actually improve the player’s odds. For example, a single deck blackjack game versus a game with two decks or more gives 0.48% back to the player. That’s not nearly as significant as the 1.39%, so don’t fall for the trap of playing a single deck game with a 6 to 5 payout on a natural blackjack.

Most other rules variations have smaller effects on the house edge, but they can add up quickly when combined. For example, if the dealer hits on soft 17 (instead of standing), the house gains 0.22%. But if the player is also limited to only being able to double down on 10 or 11, the house gains another 0.18%, for a total gain to the house of 0.4%.


The Effects of Card Counting on the Odds

Entire books have been written about counting cards and how it affects the player’s odds in the game of blackjack, but for the purposes of this page, I just want to explain how card counters get an edge over the casino. Suppose you found a way to get another 1-2% edge on the casino by raising your bets when you have better odds and lowering your bets when you have worse odds?

For example, if you had a deck with nothing but aces and tens in it, your chances of getting a natural (and the corresponding 3 to 2 payout) would be much greater, wouldn’t it? So it would make sense to bet more in that situation.

On the other hand, if all the aces in the deck have already been dealt, it’s impossible to get a natural, which means you should lower your bet.

It turns out that each card that’s dealt out of a blackjack show affects the odds by a certain amount. By tracking the ratio of high cards versus low cards that have already been dealt, a card counter can raise or lower her bets in order to take advantage of favorable situations.

By doing so, the counter can actually gain an edge over the casino. This is mostly the result of putting more money into action when the odds are good, but some card counters also make strategy adjustments based on their count, too.


Summary

This page provides an introduction to blackjack odds. Many other pages on this site go into more details about the rules variations that increase or decrease the odds in the players and the casinos’ favor. The most important things to remember are that you should always play using basic strategy, in order to minimize the house’s edge, and also to avoid playing 6:5 blackjack games.


Football Betting Odds

In sports betting, bookmakers offer odds to reflect their opinions on the probability of a result occurring. Bookmakers present odds to customers to provide them with the idea of how much they will win if placing a certain amount of money on a sports bet. On first look, odds are difficult to understand. They are probably the most complicated facet of betting to grasp for people new to the market, as they are presented in a range of different ways and with a huge spectrum of values.

Betting allows customers to guess the result of a certain event and money will be won if the match or race ends up in the way the bettor predicted. Bookmaker’s odds are the means that decides the probability of something happening – and as a result decides how much money will be won in return for predicting a sports market correctly.

A large amount of research goes into deciding odds for matches in the football market, with matters such as previous form and records considered, along with injuries, suspensions, quality of opposition and match objectives all being considered. Different bookmakers may offer slightly different odds on the same match but that is completely common, and a big part of the enjoyment of betting is finding the best potential return on a certain market.

Of course, working out how much will be made from betting a certain amount on a specific bet is now far easier to find out with the growth of online and mobile betting, with built in bet slip calculators offering the potential returns on any bet once the stake is entered; whether that be in fractional or decimal format.


The Popular Forms of Odds in the UK

While ‘American odds’ (Moneyline odds) are popular across the pond, in the United Kingdom bookmakers tend to distinguish between decimal and fractional odds only. Fractional odds are the more popular style of the two and are presented as just that; fractions. The title of decimals also speaks for itself, and most bookmakers will offer on their websites an option to switch between the two different styles to suit the user’s preference.


Fractional Odds

Fractional odds are the most common way to display betting odds in the United Kingdom and have been used most frequently in the history of British betting.

They appear in the format as, for example 2/1. The slash in between the two numbers is expressed as ‘to,’ or ‘two to one’ taking the whole number into account. The first number in the fraction will tell you how much you will win relative to the second number in the fraction. So, taking the 2/1 price as the example, a £10 bet at those odds would mean the bettor would win back two times that amount if the bet was successful. However, fractional odds don’t take into account your stake as a return, so in addition to the winnings the original stake is also added to the return. In another example, betting £10 at odds of 6/4 would mean one and a half times the original stake would be earned back in addition to the stake.


Decimal Odds

Decimal odds are the slightly less popular version of the presentation of odds and they can often be found in the settings of certain websites, if looking to change from the fractional display.

However, they are far easier to understand than the fractional type. Simply multiplying the stake by the decimal number of the betting market chosen will give the possible return amount, including the original stake. For example, betting £5 at decimal odds of 5.00, £25 will be the winning return if the bet is successful.


A Brief Overview of American Odds

American odds are also sometimes referred to as money lines, but are never used in UK betting markets.

These odds are based on a $100 stake and give both positive and negative numbers as outcomes. A positive number (+150) means that that amount of money will be won when a $100 stake is placed. However, negative odds (-150) mean that in order to win $100, the number in the bracket needs to be staked.


Odds Conversion Table

Odds in British betting markets are either presented in fractional or decimal format and can be difficult to get to grips with. Below is a table comparing the fractional and decimal equivalent of odds ranging from 100/1 to 1/100.

Betting calculators are a huge advantage nowadays with online betting, with such a wide variety of odds the table helps to put into perspective the order they each rank in.

FRACTION DECIMAL   FRACTION DECIMAL   FRACTION DECIMAL
100/1 101 13/5 3.6 9/20 1.45
80/1 81 5/2 3.5 4/9 1.444
70/1 71 12/5 3.4 2/5 1.4
66/1 67 19/8 3.375 4/11 1.364
60/1 61 23/10 3.3 7/20 1.35
50/1 51 9/4 3.25 1/3 1.333
40/1 41 11/5 3.2 3/10 1.3
33/1 34 17/8 3.125 2/7 1.286
30/1 31 21/10 3.1 1/4 1.25
28/1 29 2/1 3 2/9 1.22
25/1 26 19/10 2.9 1/5 1.2
22/1 23 15/8 2.875 2/11 1.182
20/1 21 9/5 2.8 1/6 1.167
19/1 20 7/4 2.75 2/13 1.154
18/1 19 17/10 2.7 1/7 1.143
17/1 18 13/8 2.625 2/15 1.133
16/1 17 8/5 2.6 1/8 1.125
15/1 16 3/2 2.5 2/17 1.118
14/1 15 7/5 2.4 1/9 1.111
13/1 14 11/8 2.375 1/10 1.1
12/1 13 13/10 2.3 1/11 1.0909
11/1 12 5/4 2.25 1/12 1.0833
10/1 11 6/5 2.2 1/13 1.0769
9/1 10 11/10 2.1 1/14 1.0714
9/2 9.5 21/20 2.05 1/15 1.0667
8/1 9 1/1 (evens) 2 (evens) 1/16 1.0625
15/2 8.5 20/21 1.95 1/17 1.06
7/1 8 10/11 1.909 1/18 1.0556
13/2 7.5 9/10 1.9 1/19 1.0526
6/1 7 5/6 1.833 1/20 1.05
11/2 6.5 4/5 1.8 1/22 1.0455
5/1 6 8/11 1.727 1/25 1.04
9/2 5.5 7/10 1.7 1/28 1.0357
4/1 5 2/3 1.667 1/30 1.0333
18/5 4.6 5/8 1.625 1/33 1.0303
7/2 4.5 8/13 1.615 1/40 1.025
10/3 4.333 3/5 1.6 1/50 1.02
16/5 4.2 4/7 1.571 1/60 1.0167
3/1 4 8/15 1.533 1/66 1.0152
14/5 3.8 1/2 1.5 1/80 1.0125
11/4 3.75 8/17 1.47 1/100 1.01