In previous posts we saw how to predict the outcome of sports events such as football (Football Predictions Overview and Football Predictions with Poisson Distribution) or basketball (Basketball Predictions Overview). Sports betting is a fun option to test those models, and as long practiced in a “healthy” manner, is an additional form of entertainment for sports lovers. Bookmakers allow placing bets on sports events and pay out the winnings. They offer betting lines with agreed-upon odds figures that translate in both probability of the outcome and payout for a winning bet. In this article we will see what’s the relationship between odds and probability. Furthermore, we will explore the different formats bookmakers use to show odds and learn how to convert from one to another in Excel.
Bookmakers
A bookmaker
or bookie is a private company or a person that accepts and pays out bets on
sporting and other events at agreed-upon figures (odds). Nowadays we usually
see them as online betting portals, but there are still some physical betting
“shops” too. It is not a secret that many of them are based in Gibraltar or
Malta, among other countries. The list below shows some of the most popular bookmakers (but there are many
more).
- bet365
- 1XBET
- Bwin
- VBET
- 888Sport
- BETWINNER
- UNIBET
- WilliamHILL
- betfair
- betsson
- sportingbet
- 188BET
Some of these
bookmakers offer a sign-up bonus and that kind of incentives to attract users.
Be aware that some countries do not allow international bookmakers to operate,
and online access is restricted. Such countries may have their own local bookmakers.
Some countries may have stronger regulations on gambling or even consider it
illegal.
Some
bookmakers offer more competitive odds that others, so it’s good to compare among
them before signing up. Also, some bookmakers charge a fee on winning bets that
is deducted from the profit. Let’s see now what odds mean and how they are
calculated.
Odds vs Probability
In mathematics, odds provide a measure of the likelihood of a particular outcome. In relation to probability, odds represent the probability that the outcome occurs (p) against the probability that the outcome does not occur (1-p). That’s basically shown with the following formula.
The
probability (p) is a number between 0 and 1 (also represented as percentage).
The probability of an event is the number of possibilities for that outcome
divided by the total number of possibilities. For example, when tossing a coin,
there is a 50% or 0.5 probability to get heads, i.e., one possibility or side
of the coin divided by 2 sides (total possibilities). Let’s see another
example. The probability of getting a 6 when rolling a die is 1 divided by 6
(0.1666 or 16.66%). And that’s the exact same probability of rolling any of the
other numbers.
Now let’s put
that in the form of odds. Using the formula above, we can calculate the odds
for a given probability, and vice-versa. The odds of rolling a 6 is 1 to 5 (1
side of the die is number 6 while the other 5 sides correspond to numbers 1, 2,
3, 4, 5). That’s equivalent to 1/5 or 0.2 returned from the formula above for p
= 0.1666. The odds of rolling either a 5 or a 6 are 2 to 4, rolling 4, 5, or 6 are
3 to 3 (or a 50/50 chance), and so on.
In sports
betting, odds are used (instead of probability) because they are directly
related to the winning margin of an event; they represent the potential
earnings. For example, if we could bet for number 6 when rolling a die (1 to 5
odds of success), we could earn 5 times the amount at stake (e.g., for 100 EUR
we could earn 500 EUR). While, if we could bet for just the opposite, i.e., any
number other than 6 (odds 5 to 1), we would earn only 1/5 of the amount at
stake (e.g., for 100 EUR we could earn 20 EUR). But that’s not really how it
works, bookmakers usually show odds in a different way. The most common formats
used by bookmakers are decimal odds, fractional odds, and American odds (money
line odds). The table below shows the relationship among different odds formats,
the implied probability, and the potential payout/earnings.
Most
bookmakers allow to see odds in different formats, so use the one you are
familiar with and forget about the other.
But in case you want to understand the difference among them, continue
reading to see how each format works.
Decimal odds
This is the common format used in Europe (except UK and Ireland), Australia and New Zeeland, Canada, and Singapore, among other countries. Decimal odds represent the inverse of the probability of an event (and vice-versa). For example, a 50/50 chance to win a tennis match corresponds to a decimal odd of 2. We can see that in the table above but here’s how is calculated. A 50/50 chance means there is a 0.5 or 50% probability of winning. Decimal odds are calculated as the inverse of the probability as shown below.
Similarly, we
can calculate the probability of an event based on decimal odds.
Therefore,
the higher the decimal odds, the lower the probability of the outcome.
Similarly, the lower the decimal odd, the higher the probability. For an event
with two possible outcomes such as a Tennis match, or Over/Under events, etc.,
the most probable outcome corresponds to the lowest decimal odd, which is
always below 2. For example, odds for the tennis match between Nadal and
Hanfmann could look as follows.
Nadal 1.33
Hanfmann 3.86
In terms of
probability, that means that there is a higher probability for Nadal to win the
match. Actually, based on those odds, there is around 75% winning probability
for Nadal, while only 25% for Hanfmann. Well, those are just approximate values,
the exact values are 0.7518 (75.2%) and 0.2591 (25.9%), which “surprisingly” add to about 101.1% instead of 100%. This was only mentioned as a curiosity. It is
known as over-round (bookmaker margin) and will be explained in another article.
And, what’s
they payout? Decimal odds quote the ratio of the payout amount. The payout
amount can be simply calculated as the product of the decimal number and the
betting stake. For example, if we bet 100 EUR on the winning player in the first
example above, the payout would be 2 x 100 EUR = 200 EUR. The net profit is
that amount minus the stake (200 – 100 = 100 EUR net profit). And for the other
example above (Nadal vs Hanfmann), betting 100 EUR for Nadal could bring 133
EUR (33 EUR profit) if he wins, while 100 EUR for Hanfmann would bring 385 EUR
(285 EUR profit) in case he wins.
Decimal odds
are easy to work with because they allow to quickly calculate payout earnings.
They are also convenient when placing multiple bets, as the earnings are simply
the product of the odds for each outcome wagered on. See an example of multiple
betting and payout calculations below.
Game1 PlayerA
(2) -> betting 100 EUR can bring 200 EUR (100 profit)
Game2 PlayerB
(1.5) -> betting 100 EUR could bring 150 EUR (50 profit)
Game3 …
Total = 2 *
100 + 1.5 * 100 + … = 350 EUR (350 – 200
= 150 profit)
Fractional
odds
This is the format preferred in the UK and Ireland, also quite popular in horse-racing. Fractional odds also have a straightforward relationship with probability. In a way we could say that the denominator of the fraction represents the possibilities the outcome has to occur with respect to the total number of possibilities (numerator + denominator). The probability can be computed as follows.
For example,
a fractional odd of 1/1 represents a 50/50 chance or 50% probability the event
occurs. There is 1 possibility it occurs and 1 possibility it doesn’t. The
probability for such scenario is calculated below.
A fractional odd of 2/1 for player A means the player has one possibility to win against twice that chance to lose, so to speak. Using the formula above demonstrates that the probability of player A winning is 0.333 (33.3%).
Fractional odds below 1/1 (e.g., 1/2, 1/3, 1/4, etc) correspond to higher probabilities. For example, an event with a fractional odd of 1/4 has four times higher chance to occur against one to not occur. Using the formula above to compute the probability returns 0.8 or 80% chance for that event.
Fractional odds quote the net profit to be paid relative to the stake. This means that the net profit equals the result of the fraction division times the stake. For example, betting 100 EUR on an event that is given a fractional odd of 2/1 could bring 200 EUR net profit (300 EUR total payout).
For 3/1 betting 100 EUR could bring a net profit of 300 EUR (400 EUR total), 4/1 would bring 400 EUR net, and so on. For 1/1 the net profit is a 100%, i.e., betting 100 EUR could bring another 100 EUR in profit (200 EUR total). For odds below 1/1 (e.g., 1/2, 1/3, 1/4, etc), is exactly the same, i.e., the net profit is the result of the division of the fraction multiplied by the stake. For example, betting 100 EUR on player A to win with odds of 1/2 could bring 50 EUR net profit (total payout 150 EUR).
For 1/3 betting
100 EUR could bring 33.33 EUR net profit (133.33 EUR total), for 1/4 would be 25
EUR net profit, etc.
Fractional
odds can be shown slightly different sometimes. For example, some bookmakers
show the numerator and denominator of the fraction separated by a hyphen (e.g.,
4-1 instead of 4/1). Furthermore, some bookmakers show only the numerator when the
denominator is 1 (e.g., 4 for 4/1).
American odds
This is the
preferred format in the US, hence its name. They are also known as money line
odds. They can either be positive or negative and are easily convertible from
or into fractional odds. When positive, American odds correspond to a hundred
times the numerator of fractional odds. For example, 1/1 (fractional) is
equivalent to +100 (American), 2/1 is equivalent to +200, 3/1 is +300, etc. On
the other hand, when negative, they correspond to a hundred times the
denominator. For example, 1/2 (fractional) is equivalent to -200 (American),
1/3 is equivalent to -300, 1/4 is -400, etc.
American odds
have a straightforward and direct relation with net profit. Positive odds are equivalent
to the profit percentage, no need for calculations. For example, +100 means you
can earn 100% of the stake (100 USD earnings for 100 USD bet). For +200,
earnings are 200% (200 USD for 100 USD stake), +300 is 300%, etc. When American
odds are negative, the figure indicates how much money must be wagered to win
100 USD. For example, -200 indicates that a 200 USD stake will bring 100 USD
profit, -300 means 300 USD are needed to earn 100 USD, and so on. In this case
it requires a simple division. If we place 100 USD for -200, the profit is
100/200 = 50 USD. For -300 is 100/300 = 33.33 USD, and so on.
Odds Calculator
We can put
this all together to calculate odds and probability in Excel. The file below
uses simple formulas to calculate the probability and earnings from odds, and
VBA macros to convert from one format to another for the three types we’ve seen
here (decimal, fractional, American). Enter the odds in any of the formats and
the stake to calculate the potential earnings with this useful odds calculator
in Excel. When entering the odds in any of the formats, the program updates
odds for other formats. Download the file to use the odds calculator or check
the VBA code to calculate odds in Excel
We can also create conversion tables in Excel. The file below includes the table shown earlier created with simple worksheet formulas. For simplicity, it extrapolates from fractional odds to show decimal and American odds, and calculates the implied probability, payout, and profit from decimal odds.
Click Here to download Betting Odds Calculator in Excel.
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