**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

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?

- 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.