# Gambling mathematics

Too many players just walk into the casino, start gambling, and keep going without any idea of how long they will play or how much they will risk. Dividing the average loss by the average bet yields. And, their lack of understanding is due, at least in part, to the fact that they are working from a wrong perspective. This article needs additional citations for verification. Most likely you did not play the maximum bet, which on some games is required for the maximum payoff. Betting systems votaries are spiritually akin to the proponents of perpetual motion machines, butting their heads against the second law of thermodynamics.

## Recommended online casinos

Some casino games have a skill element, where the player makes decisions; such games are called "random with a tactical element. For more examples see Advantage gambling. The player's disadvantage is a result of the casino not paying winning wagers according to the game's "true odds", which are the payouts that would be expected considering the odds of a wager either winning or losing.

However, the casino may only pay 4 times the amount wagered for a winning wager. The house edge HE or vigorish is defined as the casino profit expressed as a percentage of the player's original bet. In games such as Blackjack or Spanish 21 , the final bet may be several times the original bet, if the player doubles or splits. In American Roulette , there are two zeroes and 36 non-zero numbers 18 red and 18 black. Therefore, the house edge is 5. The house edge of casino games vary greatly with the game.

The calculation of the Roulette house edge was a trivial exercise; for other games, this is not usually the case. In games which have a skill element, such as Blackjack or Spanish 21 , the house edge is defined as the house advantage from optimal play without the use of advanced techniques such as card counting or shuffle tracking , on the first hand of the shoe the container that holds the cards. The set of the optimal plays for all possible hands is known as "basic strategy" and is highly dependent on the specific rules, and even the number of decks used.

Good Blackjack and Spanish 21 games have house edges below 0. Online slot games often have a published Return to Player RTP percentage that determines the theoretical house edge. Some software developers choose to publish the RTP of their slot games while others do not.

The luck factor in a casino game is quantified using standard deviation SD. The standard deviation of a simple game like Roulette can be simply calculated because of the binomial distribution of successes assuming a result of 1 unit for a win, and 0 units for a loss.

Furthermore, if we flat bet at 10 units per round instead of 1 unit, the range of possible outcomes increases 10 fold. After enough large number of rounds the theoretical distribution of the total win converges to the normal distribution , giving a good possibility to forecast the possible win or loss. The 3 sigma range is six times the standard deviation: There is still a ca. The standard deviation for the even-money Roulette bet is one of the lowest out of all casinos games.

Most games, particularly slots, have extremely high standard deviations. As the size of the potential payouts increase, so does the standard deviation. Unfortunately, the above considerations for small numbers of rounds are incorrect, because the distribution is far from normal.

Moreover, the results of more volatile games usually converge to the normal distribution much more slowly, therefore much more huge number of rounds are required for that. As the number of rounds increases, eventually, the expected loss will exceed the standard deviation, many times over. From the formula, we can see the standard deviation is proportional to the square root of the number of rounds played, while the expected loss is proportional to the number of rounds played.

As the number of rounds increases, the expected loss increases at a much faster rate. This is why it is practically impossible for a gambler to win in the long term if they don't have an edge.

Understanding range may lead you to wonder how most students scored. In other words, you know what they scored, but maybe you want to know about where the majority of student scores fell — in other words, the variance of scores.

The Normal Curve tells us that numerical data will be distributed in a pattern around an average the center line. Standard deviation is considered the most useful index of variability. It is a single number that tells us the variability, or spread, of a distribution group of scores.

Standard Deviation is calculated by:. The formula returns the number, in this case, 8. Since 1 SD in our example is 8. To get this range, I simply added 1 SD 8. In other words, To get that number, I took the percentages between -3 SD and 0 on the left, which equal 50 , then added the percentage from 0 to 1 SD on the right which is.

When you get to a point where you have doubled one ticket take your money and divide it so that you always have three tickets or four if that is how you want to play. A good rule of thumb is to pull any ticket that you have doubled in value. As long as you have at least one ticket with double value or greater you can risk playing down another ticket. Whether you are playing the slots, keno, or roulette you will see the temptation to play a game that promises a crazy payoff.

Always remember that the money for those payoffs comes from the players. So if you are playing the slots, pick the game that pays 4, max credits instead of the game that pays 10, max credits. Your chances of winning are better with the smaller payoffs. This is due to a low variance in the game design. If you ignore all warnings from experienced gamblers to play keno, go for a small number of picks. The safest range of picks is 3 to 5.

That is terrible advice. The faster you lose your money the sooner you will be done gambling. If you just want to get out of the casino then leave immediately and save your cash. Otherwise, stretch it out by making small bets. If the prizes are proportionally the same regardless of the size of the bet then you gain nothing by betting more.

Some people sneer at the idea of playing small wagers. That is always true. What you have to think about is how many more wagers you can make with your current bankroll, assuming you never win until the last wager. Yes, there is always that occasional event where something unexpected happens. The same principle holds true in table games and slot games. The chances of winning are less than if they are dealt an And you can always hit on an 11 or less.

But the truth is that most people lose on a wager. The house edge means nothing on any individual wager. While that seems like a great ratio it is only relevant to a large number of wagers made over a long period of time measured in months or years. They want you to pick games that are less likely to take your money through the basic rules. Although some slot machines have a theoretical return to player that approaches the RTP of blackjack and baccarat most slot machines still have worse percentages.

The best surrender rule is the early surrender, where you can fold before the dealer checks for a natural; unfortunately, most casinos now seem to favor the Late Surrender, where the dealer takes your bet if he has a natural.

You still have to make good choices. Those rules are there to give the house an advantage. A wagering option may only exist because the casino knows someone will try it despite the low chances of winning. The higher the promise of payback the less likely you will win.

The simpler bets work the same way for everyone. This idea comes from too many people playing roulette the wrong way. The odds on a single number bet are 35 to 1.