The probability of an event is a number from 0 to 1 that measures the chance that an event will occur. In this lesson, we will look into experimental probability and theoretical probability.
How to find the Experimental Probability of an event?
The Theoretical Probability of an event is the number of ways the event can occur (favorable outcomes) divided by the number of total outcomes.
What is the Theoretical Probability formula?
![Roulette probability formula Roulette probability formula](/uploads/1/2/5/2/125264324/618737764.jpg)
Record the color and return the marble.
Repeat a few times (maybe 10 times).
Count the number of times a blue marble was picked (Suppose it is 6).
The experimental probability of getting a blue marble from the bag is
How to find and use experimental probability?
The following video gives another example of experimental probability.
Example:
The spinner below shows 10 equally sized slices. Heather spun 50 times and got the following results.
a) From Heather's' results, compute the experimental probability of landing on yellow.
b) Assuming that the spinner is fair, compute the theoretical probability of landing in yellow.
Example:
According to theoretical probability, how many times can we expect to land on each color in a spinner, if we take 16 spins?
Conduct the experiment to get the experimental probability.
We will then compare the Theoretical Probability and the Experimental Probability.
Examples:
1. A spinner is divided into eight equal sectors, numbered 1 through 8.
a) What is the probability of spinning an odd numbers?
b) What is the probability of spinning a number divisible by 4?
b) What is the probability of spinning a number less than 3?
2. A spinner is divided into eight equal sectors, numbered 1 through 8.
a) What is the probability of spinning a 2?
b) What is the probability of spinning a number from 1 to 4?
b) What is the probability of spinning a number divisible by 2?
You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
![Winning Winning](/uploads/1/2/5/2/125264324/516571619.jpg)
Written by: Michael Shackleford
It turns out the probability of you winning with a straight up bet on number 16 is 2.70%. The formula can be used to calculate the probability of winning of all roulette bets. Let’s provide another example with an even-money bet. Assume you want to place an even-money bet on Red. Mar 19, 2016 Theoretical Win and Expected Value. The average of results is determined by the theoretical win formula. Hence, 50% of all results will achieve this result. For example: A split wager on single-zero roulette pays 17:1. The probability of winning a split bet is 2/37 (2 numbers wagered on from a total of 37 numbers).
The following table highlights the difference between Experimental Probability and Theoretical Probability. Scroll down the page for more examples and solutions.How to find the Experimental Probability of an event?
Confusion about Win Rate. There are all kinds of percentages in the world of gaming. Win percentage, theoretical win percentage, hold percentage, and house advantage come to mind. Sometimes casino bosses use these percentages interchangeably, as if they are just different names for the same thing. The odds of any particular number winning in roulette could be simply displayed as 1:36 or 1/36 where 36 is, once again, the number of ways to lose. Sometimes, when it comes to expressing the odds of a particular bet in roulette, they would be in reverse, indicating the odds against winning.
Step 1: Conduct an experiment and record the number of times the event occurs and the number of times the activity is performed.
Step 2: Divide the two numbers to obtain the Experimental Probability
How to find the Theoretical Probability of an event?Step 2: Divide the two numbers to obtain the Experimental Probability
The Theoretical Probability of an event is the number of ways the event can occur (favorable outcomes) divided by the number of total outcomes.
What is the Theoretical Probability formula?
The formula for theoretical probability of an event is
Experimental Probability
One way to find the probability of an event is to conduct an experiment.Example:
A bag contains 10 red marbles, 8 blue marbles and 2 yellow marbles. Find the experimental probability of getting a blue marble.
![Roulette probability formula Roulette probability formula](/uploads/1/2/5/2/125264324/618737764.jpg)
Solution:
Take a marble from the bag.Record the color and return the marble.
Repeat a few times (maybe 10 times).
Count the number of times a blue marble was picked (Suppose it is 6).
The experimental probability of getting a blue marble from the bag is
How to find and use experimental probability?
The following video gives another example of experimental probability.
- Show Step-by-step Solutions
Example:
The spinner below shows 10 equally sized slices. Heather spun 50 times and got the following results.
a) From Heather's' results, compute the experimental probability of landing on yellow.
b) Assuming that the spinner is fair, compute the theoretical probability of landing in yellow.
Theoretical Probability
We can also find the theoretical probability of an event.
Example:
A bag contains 10 red marbles, 8 blue marbles and 2 yellow marbles. Find the theoretical probability of getting a blue marble.
Solution:
There are 8 blue marbles. Therefore, the number of favorable outcomes = 8.
There are a total of 20 marbles. Therefore, the number of total outcomes = 20
Example:
Find the probability of rolling an even number when you roll a die containing the numbers 1-6. Express the probability as a fraction, decimal, ratio and percent.
Solution:
The possible even numbers are 2, 4, 6. Number of favorable outcomes = 3.
Total number of outcomes = 6
The probability = (fraction) = 0.5 (decimal) = 1:2 (ratio) = 50% (percent) Comparing theoretical and experimental probability
The following video gives an example of theoretical and experimental probability.Example:
According to theoretical probability, how many times can we expect to land on each color in a spinner, if we take 16 spins?
Conduct the experiment to get the experimental probability.
We will then compare the Theoretical Probability and the Experimental Probability.
- Show Step-by-step Solutions
Examples:
1. A spinner is divided into eight equal sectors, numbered 1 through 8.
a) What is the probability of spinning an odd numbers?
b) What is the probability of spinning a number divisible by 4?
b) What is the probability of spinning a number less than 3?
2. A spinner is divided into eight equal sectors, numbered 1 through 8.
a) What is the probability of spinning a 2?
b) What is the probability of spinning a number from 1 to 4?
b) What is the probability of spinning a number divisible by 2?
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You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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Introduction
The following table shows the house edge of most casino games. For games partially of skill perfect play is assumed. See below the table for a definition of the house edge.
Casino Game House Edge
Game | Bet/Rules | House Edge | Standard Deviation |
---|---|---|---|
Baccarat | Banker | 1.06% | 0.93 |
Player | 1.24% | 0.95 | |
Tie | 14.36% | 2.64 | |
Big Six | $1 | 11.11% | 0.99 |
$2 | 16.67% | 1.34 | |
$5 | 22.22% | 2.02 | |
$10 | 18.52% | 2.88 | |
$20 | 22.22% | 3.97 | |
Joker/Logo | 24.07% | 5.35 | |
Bonus Six | No insurance | 10.42% | 5.79 |
With insurance | 23.83% | 6.51 | |
Blackjacka | Liberal Vegas rules | 0.28% | 1.15 |
Caribbean Stud Poker | 5.22% | 2.24 | |
Casino War | Go to war on ties | 2.88% | 1.05 |
Surrender on ties | 3.70% | 0.94 | |
Bet on tie | 18.65% | 8.32 | |
Catch a Wave | 0.50% | d | |
Craps | Pass/Come | 1.41% | 1.00 |
Don't pass/don't come | 1.36% | 0.99 | |
Odds — 4 or 10 | 0.00% | 1.41 | |
Odds — 5 or 9 | 0.00% | 1.22 | |
Odds — 6 or 8 | 0.00% | 1.10 | |
Field (2:1 on 12) | 5.56% | 1.08 | |
Field (3:1 on 12) | 2.78% | 1.14 | |
Any craps | 11.11% | 2.51 | |
Big 6,8 | 9.09% | 1.00 | |
Hard 4,10 | 11.11% | 2.51 | |
Hard 6,8 | 9.09% | 2.87 | |
Place 6,8 | 1.52% | 1.08 | |
Place 5,9 | 4.00% | 1.18 | |
Place 4,10 | 6.67% | 1.32 | |
Place (to lose) 4,10 | 3.03% | 0.69 | |
2, 12, & all hard hops | 13.89% | 5.09 | |
3, 11, & all easy hops | 11.11% | 3.66 | |
Any seven | 16.67% | 1.86 | |
Double Down Stud | 2.67% | 2.97 | |
Heads Up Hold 'Em | Blind pay table #1 (500-50-10-8-5) | 2.36% | 4.56 |
Keno | 25%-29% | 1.30-46.04 | |
Let it Ride | 3.51% | 5.17 | |
Pai Gowc | 1.50% | 0.75 | |
Pai Gow Pokerc | 1.46% | 0.75 | |
Pick ’em Poker | 0% - 10% | 3.87 | |
Red Dog | Six decks | 2.80% | 1.60 |
Roulette | Single Zero | 2.70% | e |
Double Zero | 5.26% | e | |
Sic-Bo | 2.78%-33.33% | e | |
Slot Machines | 2%-15%f | 8.74g | |
Spanish 21 | Dealer hits soft 17 | 0.76% | d |
Dealer stands on soft 17 | 0.40% | d | |
Super Fun 21 | 0.94% | d | |
Three Card Poker | Pairplus | 7.28% | 2.85 |
Ante & play | 3.37% | 1.64 | |
Video Poker | Jacks or Better (Full Pay) | 0.46% | 4.42 |
Wild Hold ’em Fold ’em | 6.86% | d |
Notes
a | Liberal Vegas Strip rules: Dealer stands on soft 17, player may double on any two cards, player may double after splitting, resplit aces, late surrender. |
b | Las Vegas single deck rules are dealer hits on soft 17, player may double on any two cards, player may not double after splitting, one card to split aces, no surrender. |
c | Assuming player plays the house way, playing one on one against dealer, and half of bets made are as banker. |
d | Yet to be determined. |
e | Standard deviation depends on bet made. |
f | Slot machine range is based on available returns from a major manufacturer |
g | Slot machine standard deviation based on just one machine. While this can vary, the standard deviation on slot machines are very high. |
House Edge
The house edge is defined as the ratio of the average loss to the initial bet. The house edge is not the ratio of money lost to total money wagered. In some games the beginning wager is not necessarily the ending wager. For example in blackjack, let it ride, and Caribbean stud poker, the player may increase their bet when the odds favor doing so. In these cases the additional money wagered is not figured into the denominator for the purpose of determining the house edge, thus increasing the measure of risk.
The reason that the house edge is relative to the original wager, not the average wager, is that it makes it easier for the player to estimate how much they will lose. For example if a player knows the house edge in blackjack is 0.6% he can assume that for every $10 wager original wager he makes he will lose 6 cents on the average. Most players are not going to know how much their average wager will be in games like blackjack relative to the original wager, thus any statistic based on the average wager would be difficult to apply to real life questions.
The conventional definition can be helpful for players determine how much it will cost them to play, given the information they already know. However the statistic is very biased as a measure of risk. In Caribbean stud poker, for example, the house edge is 5.22%, which is close to that of double zero roulette at 5.26%. However the ratio of average money lost to average money wagered in Caribbean stud is only 2.56%. The player only looking at the house edge may be indifferent between roulette and Caribbean stud poker, based only the house edge. If one wants to compare one game against another I believe it is better to look at the ratio of money lost to money wagered, which would show Caribbean stud poker to be a much better gamble than roulette.
Many other sources do not count ties in the house edge calculation, especially for the Don’t Pass bet in craps and the banker and player bets in baccarat. The rationale is that if a bet isn’t resolved then it should be ignored. I personally opt to include ties although I respect the other definition.
Element of Risk
For purposes of comparing one game to another I would like to propose a different measurement of risk, which I call the 'element of risk.' This measurement is defined as the average loss divided by total money bet. For bets in which the initial bet is always the final bet there would be no difference between this statistic and the house edge. Bets in which there is a difference are listed below.
Element of Risk
Game | Bet | House Edge | Element of Risk |
---|---|---|---|
Blackjack | Atlantic City rules | 0.43% | 0.38% |
Bonus 6 | No insurance | 10.42% | 5.41% |
Bonus 6 | With insurance | 23.83% | 6.42% |
Caribbean Stud Poker | 5.22% | 2.56% | |
Casino War | Go to war on ties | 2.88% | 2.68% |
Heads Up Hold 'Em | Pay Table #1 (500-50-10-8-5) | 2.36% | 0.64% |
Double Down Stud | 2.67% | 2.13% | |
Let it Ride | 3.51% | 2.85% | |
Spanish 21 | Dealer hits soft 17 | 0.76% | 0.65% |
Spanish 21 | Dealer stands on soft 17 | 0.40% | 0.30% |
Three Card Poker | Ante & play | 3.37% | 2.01% |
Wild Hold ’em Fold ’em | 6.86% | 3.23% |
Standard Deviation
The standard deviation is a measure of how volatile your bankroll will be playing a given game. This statistic is commonly used to calculate the probability that the end result of a session of a defined number of bets will be within certain bounds.
The standard deviation of the final result over n bets is the product of the standard deviation for one bet (see table) and the square root of the number of initial bets made in the session. This assumes that all bets made are of equal size. The probability that the session outcome will be within one standard deviation is 68.26%. The probability that the session outcome will be within two standard deviations is 95.46%. The probability that the session outcome will be within three standard deviations is 99.74%. The following table shows the probability that a session outcome will come within various numbers of standard deviations.
I realize that this explanation may not make much sense to someone who is not well versed in the basics of statistics. If this is the case I would recommend enriching yourself with a good introductory statistics book.
![Winning Winning](/uploads/1/2/5/2/125264324/516571619.jpg)
Standard Deviation
Number | Probability |
---|---|
0.25 | 0.1974 |
0.50 | 0.3830 |
0.75 | 0.5468 |
1.00 | 0.6826 |
1.25 | 0.7888 |
1.50 | 0.8664 |
1.75 | 0.9198 |
2.00 | 0.9546 |
2.25 | 0.9756 |
2.50 | 0.9876 |
2.75 | 0.9940 |
3.00 | 0.9974 |
3.25 | 0.9988 |
3.50 | 0.9996 |
3.75 | 0.9998 |
Hold
Although I do not mention hold percentages on my site the term is worth defining because it comes up a lot. The hold percentage is the ratio of chips the casino keeps to the total chips sold. This is generally measured over an entire shift. For example if blackjack table x takes in $1000 in the drop box and of the $1000 in chips sold the table keeps $300 of them (players walked away with the other $700) then the game's hold is 30%. If every player loses their entire purchase of chips then the hold will be 100%. It is possible for the hold to exceed 100% if players carry to the table chips purchased at another table. A mathematician alone can not determine the hold because it depends on how long the player will sit at the table and the same money circulates back and forth. There is a lot of confusion between the house edge and hold, especially among casino personnel.
Hands per Hour, House Edge for Comp Purposes
The following table shows the average hands per hour and the house edge for comp purposes various games. The house edge figures are higher than those above, because the above figures assume optimal strategy, and those below reflect player errors and average type of bet made. This table was given to me anonymously by an executive with a major Strip casino and is used for rating players.
Hands per Hour and Average House Edge
Games | Hands/Hour | House Edge |
---|---|---|
Baccarat | 72 | 1.2% |
Blackjack | 70 | 0.75% |
Big Six | 10 | 15.53% |
Craps | 48 | 1.58% |
Car. Stud | 50 | 1.46% |
Let It Ride | 52 | 2.4% |
Mini-Baccarat | 72 | 1.2% |
Midi-Baccarat | 72 | 1.2% |
Pai Gow | 30 | 1.65% |
Pai Pow Poker | 34 | 1.96% |
Roulette | 38 | 5.26% |
Single 0 Roulette | 35 | 2.59% |
Casino War | 65 | 2.87% |
Spanish 21 | 75 | 2.2% |
Sic Bo | 45 | 8% |
3 Way Action | 70 | 2.2% |
Roulette Probability Statistics
Translation
A Spanish translation of this page is available at www.eldropbox.com.
Written by: Michael Shackleford