Sports Betting Terminology and Formulas

Sports betting is a multibillion industry spreading across all continents. Betting was known to a humankind since early times, however, no much knowledge was available about probability or either about permutation and combinations until Al-Khalil ibn Ahmad al-Farahidi (718 – 786 CE) wrote the "Book of Cryptographic Messages" which contains the first use of permutations and combinations to list all possible Arabic words with and without vowels. The mathematical methods of probability arose in the investigations first of Gerolamo Cardano in "Liber de ludo aleae" ("Book on Games of Chance"), written around 1564, but not published until 1663, and then in the correspondence Pierre de Fermat and Blaise Pascal (1654) on such questions as the fair division of the stake in an interrupted game of chance. Christiaan Huygens (1657) gave a more comprehensive treatment of the subject.

Chess versus Backgammon. Herat (Afghanistan), Timurid period, 1427. Ink, colors, silver, and gold on paper.

Below, I provide the list of useful formulas for the traditional fixed odds betting and the dynamic trading on the exchange, all in one page which makes it easy accessible for the regular punters and the professional handicappers. The types of bets or betting contracts and many more will be covered here.

I am not trying popularize some compulsive habits or encourage negative betting habits, which are mostly driven by the lack of knowledge and the misuse of tools. I believe, the more knowledge is available to one on the subject, the less likely the one becomes a pray of the hustling or follow degenerative betting patterns. Here, I intentionally omitted the phrase "gambling habits" or "gambling addiction", since I don't believe gambling and probability (as science and trade) have harm to humanity, where the terminology and practice are misused by many, who lack understanding its principles.

Type of Bets (Betting Contracts)

Bet is a contract between a punter and bookmaker allowing the punter to wage a specific amount at given odds with an expectation to win a profit in future. Single is considered a vanilla type of the betting contract, which is a selection of a specific outcome of some event. Multiple bets are permutation and combination of single contracts. Multiple bets although promise the highest return in comparison with singles, but they also bear highest risk among the betting contracts.

Double bet is a most common type of the multiple bets, which consists of two selections, both of which must win to gain a return. The double is a bet where two selections are made and if the first wins, everything that would have been returned is then used as stake for second.

Treble is a multiple bet consisting of three selections, all of which must win to gain a return. The treble is a bet where three selections are made and the returns from the first are rolled over to the second, then the returns from the second rolled over to the third.

Accumulator is a bet that combines four or more selections into a single wager that gains a return only when all parts win.

It should be noted that all selections must be mutually exclusive (independent).

Multiple Bets

Bet	Selections	Doubles	Trebles	4-Folds	5-Folds	6-Folds	7-Folds	8-Folds	Total
Trixie	3	3	1						4
Yankee	4	6	4	1					11
Canadian	5	10	10	5	1				26
Heinz	6	15	20	15	6	1			57
Super Heinz	7	21	35	35	21	7	1		120
Goliath	8	28	56	70	56	28	8	1	247

• Cross Bet (up-and-down, vice-versa) – 2 selections, 2 bets: (1) one point win A, if cash, one one point win B; (2) one point win B, if cash, one point win A.
• Patent – 3 selections, 7 bets: Trixie plus 3 singles.
• Round Robin – 3 selections, 10 bets: Trixie plus 3 Cross Bets.
• Lucky 15 (Yap) – 4 selections, 15 bets: Yankee plus 4 singles.
• Flag – 4 selections, 23 bets: Yankee plus 4 singles.
• Rounder – 3 selections, 3 bets: (1) one point win A, if cash, one point double BxC; (2) one point, win B, if cash, one point double AxC; (3) one point win C, if cash, one point double AxB.
• Roundabout as Rounder except doubles are to 2-point stakes.
• Round The Clock – 3 selections, 3 bets: (1) one point win A, if cash, one point win B, if cash, one point win C; (2) one point win B, if cash, one point win C, if cash, one point win A; (3) one point win C, if cash, one point win A, if cash, one point win B.
• Liverpool Round The Clock – 3 selections, 13 bets: Trixie, Roundabout, 3 Cross Bets (double-out stakes).
• Union Jack – 9 selections, 8 trebles from the square matrix [[A B C][D E F][G H I]]: ABC, DEF, GHI, ADG, BEH, CFI, AEI, CEG.

Odds Conversion Formulas

Useful formulas to convert odds between Decimal, Fractional and American (Moneyline) formats.

Decimal to Fractional

Subtract 1 from the decimal (D) and convert to the fraction (F), then reduce to the simplest form:
$F=D-1$
Examples:
a. $1.75-1=0.75= \frac{75}{100}= \frac{3}{4}$
b. $2.20-1=1.20= \frac{120}{100}= \frac{6}{5}$

Decimal to American

To convert the decimal (D) of less than 2.00 into the moneyline (M)
$M=\frac{-100}{D-1}$
Examples:
a. $\frac{-100}{1.9091-1} = \frac{-100}{0.9091} =-109.99=-110$
b. $\frac{-100}{1.6579-1} = \frac{-100}{0.6579} =-151.98=-152$

To convert a decimal (D) of 2.00 or higher into the moneyline (M)
$M=100\times \left(D-1\right)$
Examples:
a. $100\times \left(2.00-1\right)=100\times 1=+100$
b. $100\times \left(3.35-1\right)=100\times 2.35=+235$

Fractional to Decimal

Convert the fraction (F) into a decimal (D) and add one
$D=1+\left(F\to D^{*}\right)$
Examples:
a. $\frac{3}{4}+1=0.75+1=1.75$
b. $\frac{6}{5}+1=1.20+1=2.20$

Fractional to American

Under 1/1: Divide -100 by the fraction (F) as a decimal.
$M=\frac{-100}{F}$
Examples:
a. $\frac{-100}{\frac{1}{4}}=\frac{-100}{0.25}=-400$
b. $\frac{-100}{\frac{1}{10}}=\frac{-100}{0.10}=-1000$

Over 1/1: Convert the fraction (F) to a decimal and multiply by 100
$M=100\times F$
Examples:
a. $100\times \frac{7}{4}=100\times 1.40=+140$
b. $100\times \frac{6}{5}=100\times 1.20=+120$

American to Decimal

Positive Odds ("+"): Divide the moneyline (M) by 100 and add 1
$D=1+\frac{M}{100}$
Examples:
a. $1+\frac{100}{100}=1+1.00=2.00$
b. $1+\frac{235}{100}=1+2.35=3.35$

Negative Odds ("-"): Divide 100 by an absolute value of the moneyline (M), then add this value to 1
$D=1+\frac{100}{\overline{)M}}$
Examples:
a. $1+\frac{100}{\overline{)-110}}=1+0.9091=1.9091$
b. $1+\frac{100}{\overline{)-152}}=1+0.6579=1.6579$

American to Fractional

Positive Odds ("+"): Divide the moneyline (M) by 100 and reduce to simplest form
$F=\frac{M}{100}$
Examples:
a. $\frac{120}{100}=\frac{6}{5}$
b. $\frac{250}{100}=\frac{5}{2}$

Negative Odds ("-"): Divide 100 by an absolute value of the moneyline (M) and reduce to simplest form
$F=\frac{100}{\overline{)M}}$
Examples:
a. $\frac{100}{\overline{)-120}}=\frac{5}{6}$
b. $\frac{100}{\overline{)-300}}=\frac{1}{3}$

Cash Out

On a betting exchange you can take a position on the market and afterwards close the position in such a way that your profit is guaranteed regardless of the outcome of the event. Closing a position is not necessarily done to achieve profit but it can also be performed to minimize loss. Some of the betting exchanges make the so called "cash out" option available to you with the click of a button and you do not have to go into details of the operation because the software of the exchange does everything for you. However not all exchanges offer that feature and this is where this tool comes into use. If you have taken a lay position you are required to provide its details (lay odds and amount) as well as the back odds at which you aim to close the position. Respectively if you have taken a back position you are required to provide its details and the lay odds at which you aim to close the position. In order to achieve profit the value of the back odds should be greater than the value of the lay odds. The calculator outputs the amount with which you have to perform the final operation in order to have one and the same profit or loss in all possible cases

$A_B = \frac{o_L}{o_B}{A}_L$
$A_L = \frac{o_B}{o_L}{A}_B$
$R = A_L - A_B$

where $A_B$ - the backing amount; $A_L$ - the laying amount; $o_B$ - the backing odds; $o_L$ - the laying odds; $R$ - the resulting profit or loss.

If $o_B>o_L$, hence $R>0$ and the net profit is

$P_N = \left(1-c\right)R$
where $P_N$ - the the net profit; $c$ - the commission charged.