How to play games of chance.

How to play games of chance: Learn about payout ratio and variance

Principles described in this article can be applied to almost any game of chance in which a player plays against the house. In other words, all games where a casino collects bets and pays out potential winnings. The most famous of such games are: slot machines, roulette, blackjack a baccarat. Most of you probably know that these games are somehow set to give casinos a certain advantage in the long-run. Otherwise, the owners of the casino would be losing money by running those games.

The advantage of the casino is for every game determined by the rules of the game and by the payout rules in the case of winning. If these rules are applied over many, many rounds (sometimes even hundreds of thousands of times), casino owners can be quite certain that the amount of bets collected will exceed the amount of wins payed out. The statistics work in favour of the casino in the long-run.

Despite that long-term statistics work against you, it’s quite common to come to a casino, play, win, and leave with your winnings. The main reason is that your visit to a casino consists only of several tens of game rounds. In this case, the statistics don't have enough time to converge (show up). The outcome of your casino visit is then more determined by chance (or by your luck, if you want). It’s exactly this chance which helps lucky players win and beat the statistical advantage of the casino. To increase the chances of beating the statistics, it’s very important to know two basic characteristics of each game: payout ratio and variance.

Games of chance and payout ratio (RTP)

Payout ratio (also payout percentage, RTP – return to player) of a game of chance is the long-term statistical rate of total money won divided by the total money staked. The opposite of payout ratio is the house edge. The house edge is calculated as 100% minus payout ratio. If the payout ratio is 95% then the house edge is 100% - 95% = 5%.

Payout ratio of roulette

In European Roulette, the probability of winning when betting on the black colour is 18 black numbers divided by 37 numbers total (don’t forget the zero). The payout is 2 times the bet. The payout ratio of roulette is then 2 * 18 / 37 = 0.973 = 97.3%. House edge is then 100% - 97.3% = 2.7%. The roulette game is set to have the same RTP for all kinds of bets (colour, number, etc.).

Blackjack RTP

As blackjack rules may vary from casino to casino, its expected that RTP may differ as well. But generally, a blackjack game with utilized basic strategy gives an expected RTP of around 99.5%. In live, online blackjack the expected RTP changes as the dealer deals cards from the deck. The RTP in this case is usually between 95% and 102%. This is exploited by card-counters – players which estimate the actual RTP of the blackjack deck and try to bet high if the RTP is over 100%, to make a long-term profit. On the other hand, casinos also have means to detect such card-counters and prevent them from playing further.

Payout percentage of slot machines

The RTP of slots is usually between 92% and 99%. Payout percentage of slot machines is determined by symbols on virtual reels, by payout table, and by other specific rules applied to each particular game. If you are curious which slots have the highest RTP, check this Wolfie's collection of best paying slots.

Payout percentage of a game and payout percentage of a betting system

It’s very important to realize that the payout percentage expresses expected payout from a single game round. Let’s assume that you bet $100 on roulette and win $1200. Then you continue playing and stake 12 times $100 = $1200. The RTP must be used on each game round separately. The expected statistical casino profit in this case would be:

($100 + $1200) * (100% - 97.3 %) = $1300 * 2.7 % = $35.1

Note that 97.3% is the RTP of roulette. If you continue playing with your previous winnings, then you should expect to lose more than the previously declared house edge of the game. Most of the players place bets from their previous winnings again and again.

If you want to be a smart player, you must distinguish between RTP of the game (which applies only to a single game round) and payout percentage of your betting system . Your betting system is determined by how you play during your whole stay in a casino. This includes selection of the game and its variant/settings, rules for size of the bets, and decisions about when to stop playing. Let’s define RTP of the betting system as:A ratio of overall wins borne away by winners to net losses of other players which weren’t that lucky.

Be aware that some websites swap these two definitions. Their definition of game RTP might give a false impression that players statistically lose only a small fraction of their money (equal to house edge of the game). The rest of the money should then be redistributed among winners. In reality - if players place bets from their previous wins, they lose more. The RTP of a poor betting system may drop far below 50%. Even for blackjack – a game with 99.5% RTP. As you will see, the RTP of worse betting systems is close to 0%.

The optimal betting system is one which has the same RTP as the game played. To achieve this, you must avoid placing bets from previous winnings. Theoretically, the easiest way to achieve this is to stake your whole budget in one round. Then keep betting all-in until you lose or win a satisfactory amout of money. Roulette is a very good game for this system as you can choose the odds of your bet. Let’s assume that you have $100 to play with and $900 on your bankroll will satisfy you. Placing $100 on a square (4 numbers) is almost the optimal strategy in this case. You will either leave the casino with a nice amount of money or without $100, but your chances are quite fair. The biggest disadvantage of this approach is that you will play only for a very short period of time.

 

Variance/volatility of games of chance

To put it simply, the variance of a game determines how rapidly your bankroll is changing when you play the game. When playing a game with low variance, you win small winnings quite often. In this case, your bankroll changes quite evenly (unfortunatelly usually downwards). In the case of a high-variance game, you lose in a vast majority of rounds, but when you win, you win high. Gradual drop-downs are, from time to time, replaced by a big gain.

Variance of the game can also be described by the statistical distribution of winning amounts. As winnings are usually proportional to the stake, we are talking about distribution of winings expressed as a multiple of the bet. When betting on a colour in roulette, all winnings are paid out as double of the bet. When betting on a roulette number, all winnings are paid out as 36-times the bet.

The variance of slots is a little more complicated. You can win with many winning combinations many different multiplies of bet. Due to this, it’s not that simple to describe slot variance by one number and game providers use only loose descriptions like "small", "medium", "high". We think that slot machine variance could be described better. As this topic is quite difficult, we have decided to create a separate article about it – variance of slot machines.

Variance of the game has significant impact on your chance to walk away from a casino as a winner. The rule of thumb is, "The higher the variance is, the better". The first reason is that in a game with high variance it’s easier to win the amount that satisfies you in one round. The second reason is that more rounds end up with a loss and so you don’t bet that much from your previous winnings (which has a negative impact on the RTP of your betting system). In other words, you lose more quickly. The third reason is that if you can win a higher multiple of the bet, you can bet less to win the same amount. This again reduces the amount of your bet and therefore also reduces your long-term losses.

Just to illustrate, let’s assume a game with no variance and 99% RTP. In this game, $1 bet would pay out $0.99. The outcome of each round would be determined and winning in this game would be impossible. Of course, probably no one would like this game.

We are truly wondering how underestimated the variance is compared to RTP. Many so-called experts judge the games solely on its RTP – roulette is worse than blackjack, because it has a lower RTP. That approach is so terribly wrong. Roulette, with its 36:1 paying bet on numbers is actually a much better game to win than blackjack (without card counting). Higher variance can easily beat better RTP. Every smart gambler should know this fact and learn how to work with variance in his favour.

How variance of the game influences the RTP of the betting system

Now we will demonstrate one very simple betting system. Note how the variance of the game affects the RTP of the betting system. Imagine two players: each of them came to a casino with $100. Player X bets $10 on a number in roulette. Player Y places bets on colour. Each of them leaves the casino if his bankroll exceeds $500 or if he loses all his money. We have simulated both players one milion times (using a simple software routine with a random number generator).

Player X left the casino as a winner in 14.8 % of his attempts. His average bankroll was $68 and he played 16 rounds in average. This means that the RTP of his betting system was 95.19 %.

Note: the RTP of the betting system is calculated as the ratio of net wins to net losses. Net wins are calculated as 14.8% * ($648 - $100). Net losses are calculated as $100 * (100 % - 14.8 %). Therefore (($648 - $100) * 14.8 %) / ($100 * (100 % - 14,8 %)) = 95.19 %.

Player Y managed to win in only 5.15% of his attempts. His system gave an average win of $500, an overall RTP of 21.42%. On the other hand, player Y enjoyed playing for a much longer time – in average 274 rounds.

From this example you can clearly see that variance of the game significantly affects your chances to leave the casino as a winner. The payout ratio of player X's system was 4.4 times higher than the RTP of the system used by player Y.

How bet size affects the RTP of a betting system

Win amounts in games of chance are usually calculated from the bet size. The overall payout ratio of your betting system is therefore also affected by the size of your bets. The rule of thumb is simple: The higher the size of your bets, the higher (usually) the RTP of your betting system. We assume that all the other rules of the betting system remain the same.

We will again use roulette to demonstrate the effect of bet size on betting system results. The players again come to a casino with $100 and leave if their bankroll reaches 0 or exceeds $500. Player X bets $20 and player Y bets $5. Both players bet on colour.

After simulating one milion X players we found out that 10.9 % of them managed to get to $500. This stands for 48.68% RTP and an average of 84 rounds. On the other hand, out of one milion Y players only 0.88% of them managed to win (an average of 706 rounds, RTP 3.55%). If player Y wants to leave the casino with $500, he must win 80 times more than he loses. It seems that having such a series on roulette is quite rare.

The statistics are clear in this case. When playing low bets on low variance game, you may be playing longer, but your odds to win a satisfactory amount of money significantly drop.

The opposite extreme is a combination of a high variance game with high bets. Player Z came to a casino with $100 and he decided to play all-in on roulette on a single number. The results of simulation show that from one milion Z players, only 2.71% managed to win. However each of the winners left with $3600 (RTP 97.4 %). Each of the Z players played exactly one round.

The recommended roulette betting system

A good betting system is therefore a trade off between odds of a satisfactory win and its size and duration of your stay in a casino. Which betting system should you choose if you wish to play for a reasonable amount of time while having a good chance of winning? The following table shows the results of our simulations. The variables are the size of a single bet and bet type (colour/number). Each player begins with $100 and leaves the casino with $500 or more.

The table contains the portion of players which managed to win, average size of their win, the calculated RTP of such a betting system, and the average number of rounds played.

Bet: $0.5 $1 $2 $5
Number

9,55% x $505

RTP 42,76%

3832 rounds

13,9% x $511

RTP 66,35%

1083 rounds

16,15% x $522

RTP 81,27%

292 rounds

16,7% x $555

RTP 91,21%

52 rounds

Colour

0%

0%

0,0015% x $500

0,88% x $500

RTP 0%

RTP 0%

RTP 0,006%

RTP 3,55%

7399 rounds

3698 rounds

1850 rounds

706 rounds

 

Bet: $10 $20 $50 $100
Number

14,8% x $648

RTP 95,19%

15,9 rounds

12,8% x $760

RTP 96,88%

4,73 rounds

5,33% x $1 825

RTP 97,12%

1,97 rounds

2,71% x $3 600

RTP 97,40%

1 round

Colour

5,13% x $500

RTP 21,62%

274 rounds

10,85% x $500

RTP 48,68%

84 rounds

15,94% x $500

RTP 75,85%

15 rounds

17,9% x $500

RTP 87,21%

3 rounds

As you can see, a much higher variance of bets on number can easily beat bets that are 10-times higher on colour. In addition, your stay in the casino will take much longer. Also note that when betting $2 on colour, your chances to get to $500 are close to 0. None of the one million attempts were successfull in this case.

If your goal is to have fun and have good odds of winning, we recommend you choose a game with the highest possible variance and play it with adequatelly small bets.

In the case of roulette and a $100 budget, betting $2 - $4 on your lucky number seems to be one of the best options.

Conclusions

  • Payout percentage of games of chance applies only for a single round. Your real payout percentage will be lower due to re-betting previous winnings.
  • You statistically lose on every bet. A fewer bets means a smaller statistical profit for the casino.
  • The RTP of your betting system depends on the RTP of the game, variance of the game, size of your bets, and rules when to leave and when to keep playing.
  • Variance of the game is determined by size of potential winnings (as multiplies of the bet). The higher the multiplies of the bet you are winning, the higher the variance of the game.
  • Games with high variance are usually more favourable. The advantage of significantly higher variance can easily beat the advantage of slightly higher RTP.
  • The higher the variance, the higher the chance of turning small sums of money into huge sums.
  • The bigger the bets in one round, the higher the RTP of your betting system.
  • If you play for fun, look for games with high variance and place smaller bets. A few wins should get you to the desired win, or you will lose your whole budget. Keep in mind that you are doing a tradeoff between duration of the play and the RTP of your betting system.

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