Probabilities in Sports Betting
Welcome to my section on probabilities in sports
betting. This and racetrack betting are unique sections at the
Wizard of Odds because it can not be purely analyzed with
mathematics. In other words you can never know exactly what the
house edge is for a certain bet because of the human factor involved
in sports. However we can look at past bets available and their
eventual outcome to make some pretty accurate observations. We can
also assume that the bookmaker's actions are motivated by the desire
to balance the action on both sides of a bet so that he wins
regardless of the outcome. In other words we can assume that on
average both sides of a bet are about equal. The organization of
this section will be to address various bets available and analyze
their return.
Straight Bets
Straight bets are the most popular
and in my opinion the best kind of bet of all sports bets. This is a
simple bet on the outcome of a game, relative to a point spread. The
point spread is created to add some fictional points to the underdog
team. For example suppose the Baltimore Ravens are playing the New
Orleans Saints and New Orleans is favored to win. The expert
handicappers may determine that giving Baltimore an extra 6 points
would make it a fair bet, in other words people would bet equally on
both sides. The way this bet would look in the papers or in the
casinos is New Orleans -6 vs. Baltimore. A way to remember what this
means is that New Orleans would have a fictional negative six points
to begin the game, or that New Orleans would have to overcome a six
point handicap. If instead of a number there is the letter P (for
"pick") then there is no point spread at all and you simply "pick"
who will win the game. The point spread is set in units of 1/2.
Sometimes an apostrophe symbol is used to designate a half, for
example 6' would be a spread of 6 1/2.
When making a straight bet you must usually wager $11 to win $10.
Another way of looking at is that it is an even money bet but the
loser must pay a 10% commission. For example if you wanted to win
$50 you would have to bet $55, and if you bet $50 you would get
$50*(10/11)=$45.45 if you won (subject to rounding). Assuming that
the spread is truly fair the house edge is (.5*10 - .5*11)/11 =
4.55%. In the event that the outcome of the game is exactly
on the point spread then all bets are a push. This actually happened
twice in the Super Bowl recently. In 99 Saint Louis was favored by 7
over Tennessee and the score was 23-16. In 96 Green Bay was favored
by 14 over New England and the score was 35-21.
Sometimes the Las Vegas sport books wills have promotional 5%
straight bets. In other words the player must wager $10.50 to win
$10. For example this year (2001) the Regent casino offers 5%
straight bets during certain hours on Mondays to celebrate Monday
Night Football. The house edge on 5% straight bets for a random
picker is 2.38%, ignoring ties.
Money Lines
Money lines are another popular way to
bet on sports. This is a simple bet on the true winner of a sporting
event without any point spread. Naturally one team will be favored
most of the time so if you want to bet on the favorite you must bet
more than you stand to win (laying odds). If you bet on the underdog
you stand to win more than your bet (taking odds). Let's use the
Baltimore/New Orleans from the previous section as an example again.
The money line would look something like this: New Orleans (-210)
vs. Baltimore (+170). This means that if you prefer to bet on the
favorite, New Orleans, you must wager $210 for every $100 you win if
New Orleans wins. If you prefer to bet on the underdog, Baltimore,
you win $170 for every $100 you wager if Baltimore wins.
The way a money line is established is first the bookmaker uses
his best judgment to determine the probability that the favorite
will win. For example let's assume 60%. He then converts this to a
fair money line with no house edge. If the probability is p then the
money lines are +/- 100*p/(1-p). If p is 60% then the fair money
lines would be +150 and -150. Then the bookmaker will take a
constant and add it to the amount the favorite better must bet and
subtract is from the amount the underdog better can win. A common
constant is 10 points. In this example the bookmaker would adjust
the money lines to +140 and -160. This is referred to as a 20 cent
line, referring the total line movement from the theoretical fair
line. For games with a strong favorite the numbers of points will
increase.
If the two money lines are x and y (for example x=+140 and
y=-160) then the house edge on the taking odds (or betting on the
underdog) is (x+y)/(200+x-y). The house edge on the laying odds (or
betting on the favorite) is -100*(x/y+1)/(200+x-y). In the +140/-160
example the house edge on the taking odds is 4.00% and laying odds
is 2.50%. If the player must lay odds on either team (for example
x=-105 and y=-115) then the house edge on x is
((20000/x)+x-y+200)/(x-y+400) and on y is
((100/y)*(x-y+200)+200)/(x-y+400). In this example the house edge on
x is 4.76% and on y is 4.24%.
The following tables show the house edge according to the kind of
money line (10, 15, or 20 cents) and specific money lines.
| House Edge on 10 Cent Line Money
Line Bets |
| Money Lines |
Probability
Favorite |
House Edge |
| Underdog |
Favorite |
Underdog |
Favorite |
| -105 |
-105 |
50% |
2.38% |
2.38% |
| +100 |
-110 |
51.22% |
2.44% |
2.22% |
| +105 |
-115 |
52.38% |
2.38% |
2.07% |
| +110 |
-120 |
53.49% |
2.33% |
1.94% |
| +115 |
-125 |
54.55% |
2.27% |
1.82% |
| +120 |
-130 |
55.56% |
2.22% |
1.71% |
| +125 |
-135 |
56.52% |
2.17% |
1.61% |
| +130 |
-140 |
57.45% |
2.13% |
1.52% |
| +135 |
-145 |
58.33% |
2.08% |
1.44% |
| +140 |
-150 |
59.18% |
2.04% |
1.36% |
| +145 |
-155 |
60% |
2% |
1.29% |
| +150 |
-160 |
60.78% |
1.96% |
1.23% |
| +155 |
-165 |
61.54% |
1.92% |
1.17% |
| +160 |
-170 |
62.26% |
1.89% |
1.11% |
| +165 |
-175 |
62.96% |
1.85% |
1.06% |
| +170 |
-180 |
63.64% |
1.82% |
1.01% |
| +175 |
-185 |
64.29% |
1.79% |
0.97% |
| +180 |
-190 |
64.91% |
1.75% |
0.92% |
| +185 |
-195 |
65.52% |
1.72% |
0.88% |
| +190 |
-200 |
66.1% |
1.69% |
0.85% |
| House Edge on 15 Cent Line Money
Line Bets |
| Money Lines |
Probability
Favorite |
House Edge |
| Underdog |
Favorite |
Underdog |
Favorite |
| -105 |
-110 |
50.62% |
3.59% |
3.37% |
| +100 |
-115 |
51.81% |
3.61% |
3.14% |
| +105 |
-120 |
52.94% |
3.53% |
2.94% |
| +110 |
-125 |
54.02% |
3.45% |
2.76% |
| +115 |
-130 |
55.06% |
3.37% |
2.59% |
| +120 |
-135 |
56.04% |
3.3% |
2.44% |
| +125 |
-140 |
56.99% |
3.23% |
2.3% |
| +130 |
-145 |
57.89% |
3.16% |
2.18% |
| +135 |
-150 |
58.76% |
3.09% |
2.06% |
| +140 |
-155 |
59.6% |
3.03% |
1.96% |
| +145 |
-160 |
60.4% |
2.97% |
1.86% |
| +150 |
-165 |
61.17% |
2.91% |
1.77% |
| +155 |
-170 |
61.9% |
2.86% |
1.68% |
| +160 |
-175 |
62.62% |
2.8% |
1.6% |
| +165 |
-180 |
63.3% |
2.75% |
1.53% |
| +170 |
-185 |
63.96% |
2.7% |
1.46% |
| +175 |
-190 |
64.6% |
2.65% |
1.4% |
| +180 |
-195 |
65.22% |
2.61% |
1.34% |
| +185 |
-200 |
65.81% |
2.56% |
1.28% |
| House Edge on 20 Cent Line Money
Line Bets |
| Money Lines |
Probability
Favorite |
House Edge |
| Underdog |
Favorite |
Underdog |
Favorite |
| -110 |
-110 |
50% |
4.55% |
4.55% |
| -105 |
-115 |
51.22% |
4.76% |
4.24% |
| +100 |
-120 |
52.38% |
4.76% |
3.97% |
| +105 |
-125 |
53.49% |
4.65% |
3.72% |
| +110 |
-130 |
54.55% |
4.55% |
3.5% |
| +115 |
-135 |
55.56% |
4.44% |
3.29% |
| +120 |
-140 |
56.52% |
4.35% |
3.11% |
| +125 |
-145 |
57.45% |
4.26% |
2.93% |
| +130 |
-150 |
58.33% |
4.17% |
2.78% |
| +135 |
-155 |
59.18% |
4.08% |
2.63% |
| +140 |
-160 |
60% |
4% |
2.5% |
| +145 |
-165 |
60.78% |
3.92% |
2.38% |
| +150 |
-170 |
61.54% |
3.85% |
2.26% |
| +155 |
-175 |
62.26% |
3.77% |
2.16% |
| +160 |
-180 |
62.96% |
3.7% |
2.06% |
| +165 |
-185 |
63.64% |
3.64% |
1.97% |
| +170 |
-190 |
64.29% |
3.57% |
1.88% |
| +175 |
-195 |
64.91% |
3.51% |
1.8% |
| +180 |
-200 |
65.52% |
3.45% |
1.72% |
According to Las Vegas Sun columnist Jeff Haney casinos that
offer the best 10-cent baseball line include the Las Vegas Club,
Hard Rock, Club Fortune (Henderson), Cal-Neva (Lake Tahoe),
Stardust, Luxor, Excalibur, Barbary Coast, Gold Coast, Suncoast,
Orleans, El Cortez, Jerry's Nugget, and the Rio. He lists some
others as well that break from 10 cents early. To view his entire
article click here.
Parlays
The Parlay is a way to bet on multiple
sporting events with hope for a big payoff if all of them win. All
picks are relative to the same point spread as in straight bets. If
just one event doesn't win or draw then you lose the entire bet. If
one or more event is a draw then those events are ignored. If you
win all the other events you get paid according to the number of
events that you did win. In the event all games bet on result in a
push except one or none then the entire bet becomes a push. The
following table shows the payoff according to the number of events
bet on and the corresponding house edge, assuming that the
probability of winning any given event is 50%. In addition the table
presents the house edge of making the same number of straight bets
and letting the winnings ride every time. Note that the house edge
is less on the three team parlay.
House Edge on Parlay
and
Consecutive Straight Bets |
Number
of Events |
Pays |
House
Edge
(Parlay Bets) |
House
Edge
(Straight Bets) |
| 2 |
13-5 |
10.00% |
8.88% |
| 3 |
6-1 |
12.50% |
13.03% |
| 4 |
10-1 |
31.25% |
16.98% |
| 5 |
20-1 |
34.38% |
20.75% |
| 6 |
40-1 |
35.94% |
24.36% |
The Stardust offers better returns on their parlays than the
typical pay table above. Another novelty about the Stardust is that
all point spreads end in a half point, so there is never a tie.
Other casinos that offer the same parlay bets as the Stardust are
Sam's Town, California, Fremont, Edorado, and Jokers Wild.
| Stardust Parlays |
Number
of Events |
Pays |
House
Edge
|
| 3 |
11-2 |
18.75% |
| 4 |
12-1 |
18.75% |
| 5 |
25-1 |
18.75% |
| 6 |
51-1 |
18.75% |
| 7 |
103-1 |
18.75% |
| 8 |
207-1 |
18.75% |
| 9 |
419-1 |
17.97% |
| 10 |
849-1 |
16.99% |
Teasers
A teaser is similar to the parlay bet in
that you choose multiple events and have to win them all. However
the bookmaker teases the player with a specified number of extra
points for each team bet on. Naturally the player pays for these
extra points in the form of lower payoffs than on a parlay bet.
For example the Ravens are playing the Giants and the Ravens are
favored by 3. If you bet on the Giants are part of a 6-point teaser
you will win that game if the Ravens outscore the Giants by less
then 9 points, or if the Giants win.
The following tables show the payoff and house edge according to
the number of teaser points and number of games. Note the variation
in payoffs. For example the Fiesta Casino in Las Vegas pays 8:5 for
a 3 game, 6 point, teaser, while the Suncoast casino pays 9:5 for
the same thing.
| 6 Point Teasers |
Number
of Games |
Pays |
House Edge |
| 2 |
10:11 |
9.07% |
| 2 |
10:12 |
12.68% |
| 3 |
9:5 |
7.96% |
| 3 |
8:5 |
14.54% |
| 3 |
7:5 |
21.11% |
| 4 |
3:1 |
9.26% |
| 4 |
5:2 |
20.6% |
| 5 |
9:2 |
13.89% |
| 5 |
4:1 |
21.72% |
| 6 |
7:1 |
13.56% |
| 6 |
6:1 |
24.37% |
| 7 |
9:1 |
25.43% |
| 8 |
10:1 |
43.39% |
| 6 1/2 Point Teasers |
Number
of Games |
Pays |
House Edge |
| 2 |
5:6 |
8.84% |
| 2 |
10:13 |
12.03% |
| 3 |
8:5 |
8.84% |
| 3 |
7:5 |
15.85% |
| 3 |
6:5 |
22.87% |
| 4 |
5:2 |
13.47% |
| 4 |
2:1 |
25.83% |
| 5 |
4:1 |
12.84% |
| 5 |
7:2 |
21.55% |
| 6 |
6:1 |
13.95% |
| 6 |
5:1 |
26.25% |
| 7 |
8:1 |
21.99% |
| 8 |
9:1 |
38.88% |
| 7 Point Teasers |
Number
of Games |
Pays |
House Edge |
| 2 |
5:7 |
11.23% |
| 3 |
3:2 |
6.84% |
| 3 |
6:5 |
18.02% |
| 3 |
1:1 |
25.47% |
| 4 |
2:1 |
19.55% |
| 4 |
9:5 |
24.92% |
| 5 |
7:2 |
13.16% |
| 5 |
3:1 |
22.81% |
| 6 |
5:1 |
16.68% |
| 6 |
4:1 |
30.57% |
| 7 |
7:1 |
20.06% |
| 8 |
8:1 |
35.28% |
Based on all 1975 NFL games in the 93 through 2000 seasons the
following are the probability of a win, loss, and draw according to
the number of teaser points.
| Probabilities in Professional
Football |
Number
of Events |
6 points |
6.5 points |
7 points |
| Win |
67.69% |
69.61% |
70.89% |
| Lose |
30.39% |
29.11% |
27.62% |
| Draw |
1.92% |
1.28% |
1.49% |
Stardust Teasers
The Stardust offers a 5-point
teaser where ties win. In addition they move point spreads ending
with a half by a half point, so every game has the potential to end
in a tie. The probability of any random pick winning is 67.1%, based
on the 93 through 2000 NFL seasons. Other casinos that offer the
same teaser bets as the Stardust are Sam's Town, California,
Fremont, Eldorado, and Jokers Wild.
| Stardust Teasers |
Number
of Games |
Pays |
House Edge |
| 3 |
3-2 |
24.56% |
| 4 |
3-1 |
19.04% |
| 5 |
5-1 |
18.54% |
| 6 |
7-1 |
27.15% |
| 7 |
9-1 |
38.92% |
| 8 |
14-1 |
38.55% |
| 9 |
19-1 |
45.04% |
| 10 |
29-1 |
44.70% |
| 11 |
39-1 |
50.55% |
| 12 |
59-1 |
50.24% |
| 13 |
99-1 |
44.38% |
| 14 |
149-1 |
44.04% |
| 15 |
319-1 |
19.92% |
Buying Half a Point
The straight bet player has the
option to move the point spread 1/2 point to his advantage. The cost
of this half point is laying 120, as opposed to 110. The most
oportune time to buy a half point is when one team is favored by
2.5, 3, 6.5, or 7. This is because many games end in a 3 or 7 point
difference and the extra half point can either turn a loss into a
draw or a draw into a win. However many sportbooks do not allow
purchasing a 1/2 point on these spreads for exactly this reason.
Over 1975 games during the 1993 to 2000 seasons the overall house
edge by purchasing the extra half point is 4.13%.
Futures
Propostion bets usually pay long odds
that a specified team will win the playoffs or the season
championship. For example before the beginning of the 2000 baseball
season a bet that the Baltimore Orioles winning the world series
paid 35 to 1 odds.
To calculate the house edge on a set of futures bets follow these
steps:
- Convert all winning odds to a fair probability of winnings.
For example if the Orioles paid 35 to 1 odds then a fair
probability of winning would be 1/36. In general if the odds pay x
to y then the probability of winning is y/(x+y).
- Add all the probabilities form step 1. Call this total t.
- The overall house edge on all futures bets is 1-(1/t).
Lets look at an example. The Suncoast casino in Las Vegas offered
the following odds to win the Pontiac Excitement 400, which ran on
May 5, 2001. The following table shows total probability of 155.04%.
The house edge is 1-1/1.5504=0.3550, or 35.50%.
| Odds to Win Pontiac Excitement
400 |
| Driver |
Pays |
Fair Probability |
| Dale Jarrett |
4 to 1 |
20.00% |
| Dale Earnhardt Jr. |
7 to 1 |
12.50% |
| Bobby Labonte |
7 to 1 |
12.50% |
| Tony Stewart |
6 to 1 |
14.29% |
| Mark Martin |
10 to 1 |
9.09% |
| Jeff Burton |
9 to 1 |
10.00% |
| Rusty Wallace |
5 to 1 |
16.67% |
| Field (all others) |
3 to 2 |
4.00% |
| Total |
|
155.04% |
Using this method the following list presents the house edge on
futures bets on the Superbowl XXXVI at various casinos.
| House Edge on Superbowl
Futures |
| Casino |
House Edge |
| Imperial Palace |
28.09% |
| Palms |
32.1% |
| Frontier |
32.93% |
| Venetian |
33.57% |
| Rio |
33.78% |
| MGM Grand |
34.72% |
| Bellagio |
34.72% |
| Golden Nugget |
34.72% |
| Holiday |
34.72% |
| Stratosphere |
36.64% |
| Arizona Charlie's |
36.64% |
| Aladdin |
36.82% |
| Hard Rock |
37.09% |
| Suncoast |
38.3% |
| Regent |
40.35% |
| Ballys |
41.74% |
| Paris |
41.74% |
| Flamingo |
41.74% |
| Hilton |
41.74% |
| Caesars Palace |
41.74% |
| Mandalay Bay |
42.49% |
| Santa Fe Station |
43.66% |
| Plaza |
53.36% |
| Stardust |
53.57% |
| California |
53.57% |
| Luxor |
55.41% |
| Excalibur |
56.74% |
| Monte Carlo |
56.74% |
| Tropicana |
57.1% |
| Gold Coast |
57.33% |
| Circus Circus |
57.54% |
| Harrah's |
58.41% |
| San Remo |
59.09% |
| Fitzgeralds |
61.2% |
| Lady Luck |
61.2% |
| Binion's Horseshoe |
61.6% |
| Sahara |
68.69% |
Many groups of sportsbooks share the same odds on futures with
each other. For example Ballys, the Paris, and the Flamingo Hilton
all offer the same odds, which are all owned by Park Place
Entertainment. There is not always an obvious connection among
sportbooks. For example the futures at the Rio are much more
competitive than at the Las Vegas Harrah's, despite the Rio being
owned by Harrah's. The MGM Grand offers the same odds on futures as
the Bellagio. The Monte Carlo offers the same futures as the
Excalibur.
Sources
Most of this section would not have been
possible without the archives of past point spreads and scores at
the Gold Sheet. Payoffs on
parlay and teasers were taken from various casinos in Las Vegas.
