# Poisson Calculator

### Inputs

The inputs are the expected average, which is the total of occurrences of the particular event although this must be a positive number. Also inputted is the proposition, which is the total of occurrences which the bet will specify – either a positive number or a positive number plus a half.

### Outputs

The outputs include the odds (description of the bet terms), the percentage (the chances of the bet winning) and the money line (which is the fair odds on the bet).

### Example:

There is an event that the gambler considers to be Poisson and the book has offered a prop bet on it (this could be the number of times the Knicks make 3-point attempts in one game). The line is above 15.5-105 and under 15.5-115.

The gambler might think that with historical averages taken into consideration the number attempted by the Knicks will be 15.2. So is it reasonable to go with the under bet?

### Solution:

In order to reach as solution the gambler should select ‘One Variable’ radio button, then enter the figure of 15.2 into the text box marked ‘Expected Average’. Into the text box marked ‘Proposition’ the figure of 15.5 is entered. Then simply click the ‘calculate’ button.

This will show that the probability of reaching the under bet is 54.7511%, which equates to a fair money line of -121.05. As the gambler would be putting -115 on the bet the edge is therefore positive. (To determine the level of positively – 54.7611%* 100/115 – 45.2389% = 2.3794%).

The Poisson distribution is when the proposition bets take the form of ‘How many?’ for events. This means that a person can work out the average number of times that an event has occurred over a specific period of time and then use this information to calculate the probability of this event occurring a certain number of times. An example of this is expecting a basketball player to make 12 of the 3-point attempts during one game. The Poisson distribution will calculate that the player has a 10.4837% probability of making 10 of the 3-point attempts and this increases to 42.4035% for more than 12 attempts.

A number of conditions need to be met for Poisson to apply:

• The events considered need to happen one at a time, so the total number of points scored during a basketball game will not qualify as Poisson.
• The events need to have a known average rate and occur randomly in a way which is not related to the total which occurred earlier during the time interval.
For example, a hockey team is likely to change its goalie if it is losing, so this would then deviate from the Poisson system.
• The total occurrences of this particular event would need to be proportional to the period of time for the calculation, so a game took 2 hours instead of 1 then it should be expected that the number of events would also double.
• The number of opportunities that there are for the event to occur needs to be high when compared to the likelihood of the event. (The total number of wins in a football season would not be considered Poisson).

There are several examples of Poisson events:

• The total number of times the phrase ‘Holy Cow’ is used by Phil Rizzuto during a baseball game.
• The total number of sacks by a team’s defense during a game (not just a single player).
• The total number of touches made by a football running back in the course of one game.
• The total number of technical fouls made by one team during a basketball game.
• The total number of times the telephone will ring while you are watching a football game.

In addition it is possible to compare two events which are Poisson. A bet may be offered on the proposition that a football defense might have more sacks during one game than a kicker may make field goals during another game. These events would need to be independent of each other – so the outcome of one would have no bearing on the outcome of the other. The calculator can be used to determine the probability and the fair odds associated with it for Poisson events which are both one and two variable.