# Luck Factor in a Casino Game

*Luck Factor in a Casino Game:* 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.

^{} Despite the set-theoretical RTP, almost any outcome is possible in the short term.

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### Luck Factor in a Casino Game

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.

For the binomial distribution, SD is equal to {\displaystyle {\sqrt {npq}}}.

Where {\displaystyle n} is the number of rounds played.

{\displaystyle p} is the probability of winning, and {\displaystyle q} is the probability of losing.

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## Luck Factor in a Casino Game

Furthermore, if we flat bet at 10 units per round instead of 1 unit.

The range of possible outcomes increases 10 fold.

Therefore, SD for Roulette even-money bet is equal to {\displaystyle 2b{\sqrt {npq}}}.

Where {\displaystyle b} is the flat bet per round, {\displaystyle n} is the number of rounds, {\displaystyle p=18/38}, and {\displaystyle q=20/38}.

## Luck Factor in a Casino Game

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.

For example, after 100 rounds at $1 per round.

The standard deviation of the win (equally of the loss) will be {\displaystyle 2\cdot \$1\cdot {\sqrt {100\cdot 18/38\cdot 20/38}}\approx \$9.99}.

After 100 rounds, the expected loss will be {\displaystyle 100\cdot \$1\cdot 2/38\approx \$5.26}.

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## Luck Factor in a Casino Game

The 3 sigma range is six times the standard deviation: three above the mean, and three below.

Therefore, after 100 rounds betting $1 per round.

The result will very probably be somewhere between {\displaystyle -\$5.26-3\cdot \$9.99} and {\displaystyle -\$5.26+3\cdot \$9.99}, i.e., between -$34 and $24.

There is still a ca. 1 to 400 chance that the result will be not in this range, i.e. either the win will exceed $24, or the loss will exceed $34.

## Luck Factor in a Casino Game

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.

## Luck Factor in a Casino Game

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.

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## Luck Factor in a Casino Game

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).

It is the high ratio of short-term standard deviation to expected loss that fools gamblers into thinking that they can win.

## Luck Factor in a Casino Game

The volatility index (VI) is defined as the standard deviation for one round, betting one unit.

Therefore, the VI for the even-money American Roulette bet is {\displaystyle {\sqrt {18/38\cdot 20/38}}\approx 0.499}.

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## Luck Factor in a Casino Game

The variance {\displaystyle v} is defined as the square of the VI.

Therefore, the variance of the even-money American Roulette bet is ca. 0.249.

Which is extremely low for a casino game.

The variance for Blackjack is ca. 1.2, which is still low compared to the variances of electronic gaming machines (EGMs).

## Luck Factor in a Casino Game

Additionally, the term of the volatility index based on some confidence intervals are used.

Usually, it is based on the 90% confidence interval.

The volatility index for the 90% confidence interval is ca. 1.645 times as the “usual” volatility index that relates to the ca. 68.27% confidence interval.

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## Luck Factor in a Casino Game

It is important for a casino to know both the house edge and volatility index for all of their games.

The house edge tells them what kind of profit they will make as percentage of turnover.

And the volatility index tells them how much they need in the way of cash reserves.

The mathematicians and computer programmers that do this kind of work are called gaming mathematicians and gaming analysts.

Casinos do not have in-house expertise in this field.

So, they outsource their requirements to experts in the gaming analysis field.