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Poisson Calculator

The Poisson calculator is used by bettors to find the percentage of the possible outcomes of mainly propositional or “prop” bets. This tool is great for finding and edge on the odds offered.

There are more details below on how to use the tool and when to use it.

The sportsbooks below all offer prop bets that you can use the Poisson calculator with.


Expected Average: The number of expected occurrences of the event (any positive number)
Proposition: The number of occurrences specified in the terms of the bet (any non-negative integer or integer plus a half)


Odds of: Describes bet terms
Percentage: Probability of bet winning
Money Line: Fair odds (zero-vig) on bet


Let’s say a book is offering up a prop bet on an event you to believe to be Poisson — let’s say the number of 3-point attempts made by the Knicks in a particular game. The line is over 15.5 -105 under 15.5 -115.
You think that the based on historical averages the expected number of Knicks 3pt attempts is actually 15.2. Is the under bet positive expectation?
Solution:Select “One Variable” radio button
Enter 15.2 into “Expected Average” text box
Enter 15.5 into “Proposition” text box
Click “Calculate”
We see that the probability of hitting the under (“Less than 15½”) is 54.7611%, corresponding to a fair money line of -121.05
Since you’d only be laying -115 on the bet your edge would be positive. (How positive? Your edge would be 54.7611% * 100/115 – 45.2389% = 2.3794%)
The events underlying certain proposition bets of the form “How many … ?” follow what is known as the Poisson distribution. If an event is Poisson then it has the property that if you know the average number of times it’s expected to occur over a given time interval, then you can estimate the probability of the event occurring any number of times. (For example if you expect a basketball player will make 12 three-point attempts in a given game then the Poisson distribution tells us that the player makes exactly 10 3-point attempts during the game will be roughly 10.4837% , and the probability that he’ll make more than 12 attempts is roughly 42.4035%). For an event to be Poisson, these conditions need to be met:the event needs to occur one at a time (so the number of points scored in an basketball game couldn’t be Poisson);
the event needs to occur randomly but at a known average rate that is unrelated to the number of occurrences earlier in the time interval (so one way the number of goals scored in a hockey game would deviate from Poisson would be insofar as a team would be likely to eventually pull its goalie if it’s losing);
the number of occurrences of the event needs to be proportional to the time period (meaning that if a game were twice as long, we’d expect the event to occur twice as many times); and
the number of opportunities for the event occurring need to be very large relative to the likelihood of the event (so the number of wins in a football season couldn’t be Poisson).
Examples of (approximately) Poisson events include:

the number of times Phil Rizzuto says “Holy Cow” during a baseball broadcast
the number of total sacks by a football defense (although not by a single player) in a game;
the number of touches by a football running back in a game;
the number of technical fouls by a basketball team in a game;
the number of phone calls you receive during a Sunday afternoon football game.

It’s also possible to compare two Poisson events of the form “How many … versus How many … ?”. For example, a book might offer the proposition that a defensive line might have more sacks in one game plus three than a kicker might have field goal attempts in another game. To be able to use the Poisson distribution to compare these two events the events need to be independent meaning that knowing the outcome of one event tells you nothing about the likelihood of the outcome of the other. The Poisson calculator calculates the probability and associated fair odds of both one-variable and two-variable Poisson events.

Poisson Calculator

Odds of Percentage Money Line
More Than
Less Than
At Least
At Most
A = B
A > B
A < B
A = B
A = B

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Poisson Distribution Calculator

Poisson distribution helps bettors find out the likelihood of an event happening within a particular period of time. It is often used to calculate the chances of a prop bet paying off. This poisson distribution calculator does all the hard work for you, quickly providing you with the probability of an event occurring within a game.

What is Poisson Distribution?

The concept of poisson distribution can be applied to any walk of life. Retailers can use it to determine the number of customers that might visit a store on a particular day, while meteorologists can use it to predict the chances of a storm hitting an area during a given month. In sports betting, poisson distribution is generally used to calculate the probability of a prop bet paying off in a game or a specific period (try it out at our prop betting sites). It essentially helps bettors predict the probability of certain events happening in a fixed interval, but you need to know the historical average rate of success before using the probability calculator.

What is the Poisson Distribution Formula?

You can only conduct a poisson probability experiment if you know the average number of times an event occurs within a time interval. This average rate of success number is represented by ?, the Greek letter lambda. You can think of ? as the mean. The desired number of times the that you want the event to occur is symbolized by x. This is known as the Poisson random variable. The poisson probability mass function is: P(X=k)=k!?ke??, where e is Euler’s number.

You can use this formula to determine the Poisson probability and cumulative probability of a number of events occuring, entering ? based on the historical average and x based on the prop bet being offered. It looks extremely convoluted, and you can trawl through several pages of information to learn more about x e ?, ? x, x x, ? ?, e ? and so on. The formula is broken down into many different variants for specific purposes. Statisticians might enjoy examining probability mass f ( x , ? ) = e ? ? ? x ? ( x + 1 ) ( 2 ) lower cumulative functions, ( x , ? ) = e ? ? ? x? ( x + 1 ) ( 2 ), ^3x is e 3 x e 3 x , and e^(3x) is e 3 x e 3 x, ? ? x x ! , x = 1 , 2 , 3 , … P ( X = x ) = e ? ? ? x x ! , x = 1 , 2 , 3, P ( X = x ) = e ? ? ? x x ! , x = 1 , 2 , 3 , … P ( X = x ) = e ? ? ? x and so on, but that sort of thing will not mean much to the average sports bettor.

Fortunately, you do not actually need to have a thorough grasp of those formulae. You can simply use our poisson calculator to work out the number of times an event will occur in a given time interval. It uses simple language that all sports bettors can understand, allowing you to quickly decide whether to place a prop bet from a position of strength, but it is underpinned by advanced statistical calculation methods, ensuring you receive a totally accurate probability result each time.

How to Use Poisson Calculator

Follow these steps at one of our best sports betting sites to work out the poisson probability distribution rate using this probability poisson distribution calculator:

• Choose either one variable or two variables.• Enter the Expected Average. This is ?, the historical average rate at which an event has occurred, such as a basketball team scoring 18.2 points in the second quarter or a soccer team scoring 2.4 goals per game.• Enter the Proposition. This is x, the random variable, or desired number of times you want the event to occur. For example, if the total points line was set at 37.5 in an NFL game, you would enter 37.5.• Click on “Calculate”.

The poisson distribution calculator will then tell you the probability of the over or the under paying off, based on the historical average rate. The probability results are divided into three sections: “Odds of”, “Percentage” and “Moneyline”. “Odds of” describes the terms of the bet (over or under). The percentage represents the chances of the over or the under paying off in that time interval, based on the historical probability rate of success. Moneyline tells you the fair odds of the bet, based on zero vig. That allows you to determine whether there is a poisson opportunity for you to capitalize upon.

An Example

Let’s say a sportsbook is offering a prop bet on the amount of three-point attempts the New York Knicks will make in an NBA game. It has -105 on over 15.5 attempts, and -115 on under 15.5 attempts. You have found that the Knicks average 15.2 three-point attempts per game. You now have the random variable (the prop bet) and the average rate of success, so you can work out the poisson probability (p x) using the poisson distribution calculation tool:

• Select “One Variable”• Enter 15.2 into “Expected Average” text box• Enter 15.5 into “Proposition” text box• Click “Calculate”

You will then see that more than 15.5 has an average percentage of 45.24%, while less than 15.5 has an average percentage of 54.7%. The Moneyline odds with zero vig on over 15.5 would therefore be +121.05, and the true Moneyline odds on under 15.5 would be -121.05. You would then be able to see that -115 is an attractive price on under 15.5, as the true odds should be -121.05 based on the average number of times the event has occurred in the past. This poisson probability distribution analysis helps you make educated betting decisions from a position of strength.

Who Should Use This Poisson Distribution Calculator

You should use this calculator for poisson distribution probability if you want to bring a more rigorous, mathematical approach to prop betting. It can be used to help you decide whether to bet over or under on total points, yards, goals, attempts, corners, free-kicks, three-pointers and so on. You can use the poisson’s distribution calculator to work out the likely success rate of team and player props, include pre-match wagers, live betting and futures betting. You just need the random variable and the average number of times the event has occurred in order to arrive at the poisson probability, also known as p x.

Anyone can benefit from taking the time to calculate poisson distribution, from novice handicappers to veteran bettors. When used correctly, the poisson’s distribution calculator should help you win more prop bets over the course of the year, because you will be armed with the probability distribution before betting on the random variable. You should therefore bookmark this poisson distribution calculator page and use it on a regular basis to work out the probability of each wager paying off.

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