Poisson Props Notes
If an event can happen at any moment with equal probability, then the total number of events over a given period of time would follow the Poisson distribution.
Let’s consider an example. Suppose you are the mayor of a small town in northern Alaska. The ground freezes over the winter so you have to dig some empty graves before permafrost sets in to be prepared for any deaths that take place during the winter. Suppose further that on average 5 people die per winter. If you dig 8 graves to be safe, what is the probability you will run out?
In this case the number of deaths follows a Poisson distribution, because each person’s probability of dying is independent of everybody else.
The general formula for the probability that a Poisson random variable will be x, given a mean of m is
e-m * mx/x!
So in the example the following table shows the probability for 0 to 8 deaths.
|
Deaths |
Formula |
Probability |
|
0 |
exp(-5)*5^0/0! |
0.67% |
|
1 |
exp(-5)*5^1/1! |
3.37% |
|
2 |
exp(-5)*5^2/2! |
8.42% |
|
3 |
exp(-5)*5^3/3! |
14.04% |
|
4 |
exp(-5)*5^4/4! |
17.55% |
|
5 |
exp(-5)*5^5/5! |
17.55% |
|
6 |
exp(-5)*5^6/6! |
14.62% |
|
7 |
exp(-5)*5^7/7! |
10.44% |
|
8 |
exp(-5)*5^8/8! |
6.53% |
|
Total |
93.19% |
So the probability of 8 or fewer deaths is 93.19%. So the probability of running out is 100%-93.19% = 6.81%.
Excel can give both the probability that a Poisson random variable will be exactly x or x or less. The formula for exactly x is poisson(x,m,0). For example the probability of exactly 8 deaths is poisson(8,5,0) = 6.53%. The formula for 8 or fewer deaths is poisson(8,5,1) = 93.19%.
There are lots of football props that can be estimated using the Poisson distribution. The key is putting in the right mean. Let’s look at some examples taken from the Las Vegas Hilton for the 2005 Super Bowl.
Total QB sacks by both teams.
Over 4.5 -130
Under 4.5 EV
For those who may not understand the terminology –130 means the bettor must risk $130 to win $100 (plus the original wager is returned) or any proportional therein. EV means even money. We’ll see positive numbers in other examples, which is how much the player wins for a $100 bet (plus the original wager is returned). For example +150 would mean a $100 bet wins $150.
Let’s assume that for whatever reason you feel the expected number of sacks is 4.8. The following table shows the probability the total will be 4 or less.
|
Sacks |
Formula |
Probability |
|
0 |
exp(-4.8)*4.8^0/0! |
0.82% |
|
1 |
exp(-4.8)*4.8^1/1! |
3.95% |
|
2 |
exp(-4.8)*4.8^2/2! |
9.48% |
|
3 |
exp(-4.8)*4.8^3/3! |
15.17% |
|
4 |
exp(-4.8)*4.8^4/4! |
18.20% |
|
Total |
47.63% |
In Excel the probability can be found as poisson(4,4.8,1) = 47.63%.
Let’s evaluate the expected value of each bet.
The under bet has a 47.63% chance of winning and pays 1 to 1. The probability of losing is thus 100%-47.63% = 52.37%. The formula for the expected value of any bet that pays on a "for one" basis is pr(win)*pays – pr(lose). In this case 47.63%* 1 – 52.37% = -4.75%.
The over bet pays $100 for a $130 bet, or 100/130 = 0.7692 to 1. The probability of winning is the same as the probability of under bet losing (because there are no ties). So the expected value is 52.37%*0.7692 – 47.63% = -7.34%.
Let’s try another one.
Total Interceptions by Both Teams
Over 2.5 +150
Under 2.5 -180
Suppose you feel the expected number of interceptions is 1.8. The following table shows the probability of 2 or less.
|
Interceptions |
Formula |
Probability |
|
0 |
exp(-1.8)*1.8^0/0! |
16.53% |
|
1 |
exp(-1.8)*1.8^1/1! |
29.75% |
|
2 |
exp(-1.8)*1.8^2/2! |
26.78% |
|
Total |
73.06% |
At –180 the under 2.5 bet pays 1/1.8 or 0.5556 to 1. The probability of losing is 100%-73.06% = 26.94% So the expected value of the under bet is 73.06%*0.5556 – 26.94% = +13.65%. The expected value of the under bet is 26.94%*1.5 – 73.06% = -32.66%.
Here is a third prop.
Total field goals made by both teams.
Over 3.5 +120
Under 3.5 -150
You feel the average number is 2.9. The following table shows the probability there will be 3 or less.
|
Field Goals |
Formula |
Probability |
|
0 |
exp(-2.9)*2.9^0/0! |
5.50% |
|
1 |
exp(-2.9)*2.9^1/1! |
15.96% |
|
2 |
exp(-2.9)*2.9^2/2! |
23.14% |
|
3 |
exp(-2.9)*2.9^3/3! |
22.37% |
|
Total |
66.96% |
So the probability of over 3.5 field goals is 100%-66.96% = 33.04%. The under 3.5 bet pays 1/1.5 = 0.6667 to 1. The expected value of the under bet is 66.96%*0.6667 – 33.04% = 11.60%. The expected value of the over bet is 33.04% * 1.2 – 66.96% = -27.32%.
Here is a fourth prop.
Total 3rd Down Conversions by Eagles.
Over 4.5 -200
Under 4.5 +170
You feel the expected number is 5.2.
|
Conversions |
Formula |
Probability |
|
0 |
exp(-5.2)*5.2^0/0! |
0.55% |
|
1 |
exp(-5.2)*5.2^1/1! |
2.87% |
|
2 |
exp(-5.2)*5.2^2/2! |
7.46% |
|
3 |
exp(-5.2)*5.2^3/3! |
12.93% |
|
4 |
exp(-5.2)*5.2^4/4! |
16.81% |
|
Total |
40.61% |
The probability of over 4.5 is 100%-40.61% = 59.39%. The expected value of the under bet is 40.61%*1.7 – 59.39% = 9.65%. The expected value of the over bet is 59.39%*0.5 – 40.61% = -10.92%.