### Two Variable Poisson Calculator

### Calculates win probabilities and odds for Poisson-style proposition bets based upon an underlying win percentage

### Inputs

**Event A Expectation:**The number of expected occurrences of the first event (any positive number)**Event B Expectation:**The number of expected occurrences of the second event (any positive number)**Spread:**The number of occurrences of event A added to or subtracted from event A occurrence total for comparison to event B (any integer or integer plus a half)

### Outputs

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

### Example:

Let’s say a book is offering up a prop bet on a combination of events, each one of which you to believe to be Poisson and independent — let’s say the number of 3-point attempts made by the Knicks in a particular game versus the number of 3-point attempts made by the Nets in a different game. The line is Nets -1½ -110, Knicks +1½ -110.

You think that the based on historical averages the expected number of Knicks 3pt attempts is 15.2, and the expected number of Nets 3pt attempts is 17.0. Is the Nets -1½ bet positive expectation?

### Solution:

- Select “Two Variable” radio button
- Enter 15.2 into “Event A Expectation” text box
- Enter 17.0 into “Event B Expectation” text box
- Enter 1.5 into “Spread” text box
- Click “Calculate”
- We see that the probability of hitting the Nets -1½ (“A +1½ < B”) is 52.0527%, corresponding to a fair money line of -108.56
- Since you’d have to lay -110 on the bet your edge would be negative. (How negative? Your edge would be 52.0527% * 100/110 – 47.9473% = -0.6267%)

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.