Calculations

forms of odds

(a) fractional odds

British (fractional odds) - expressed as a fraction, with the numerator representing the amount of profit you will win for every unit of stake, and the denominator representing the amount of your stake

f=\frac{f_1}{f_2}

$f_1,f_2$ are two integer components that make up the displayed odds

after the division of the displayed odds $f$ is the decimal form (not to be confused with decimal odds) of the fractional odds

w=p+p \frac{\alpha}{\beta}=p\left(1+\frac{\alpha}{\beta}\right)

(b) decimal odds

European (decimal odds) - represent the amount of money you will win for every unit of money you stake (excluding stake)

d

$d$ is a decimal float that is a multiplier of your stake that results in the amount you will win

w=pd

(c) money line odds

American (money line odds) - expressed as a positive or negative number, with the positive number representing the amount of money you will win for every $100 you stake, and the negative number representing the amount of money you need to stake in order to win $100

\pm m

$m$ is a figure that shows how much you will win or how much you need to win

one odd can be expressed positively or negatively, depending on how you want to view the odds

depending on the value of the fractional odds or decimal odds or money line odds (ultimately depending on which side of even) different formulas must be used, this is common practise in the betting industry

obviously you can ignore which specific formula you “should” use and and put the value into either of the two formulas, resulting in two money line values for the corresponding one fractional value and one decimal value

graphical combination of moneyline and decimal, fractional

converting forms of odds

fractional $\leftrightarrow$ decimal

fractional odds and decimal odds are equivalent apart from the fractional form includes original stake

f=\frac{f_1}{f_2}=d-1\quad \Leftrightarrow \quad d=\frac{f_1}{f_2}+1=f+1

moneyline $\leftrightarrow$ decimal

\{1\leq d<2\cup m\leq-100\} \quad d=1-\frac{100}{m} \quad \Leftrightarrow \quad m=\frac{100}{1-d}

\{2\leq d \leq \infty \cup 100\leq m \} \quad d=1+\frac{m}{100} \quad \Leftrightarrow \quad m=100(d-1)

moneyline $\leftrightarrow$ fractional

\{0\leq f<2\cup m\leq-100\} \quad m = -\frac{100}{f} \quad \Leftrightarrow \quad f=-\frac{100}{m}

\{2\leq f \leq \infty \cup 100\leq m \}\quad m=100f \quad \Leftrightarrow \quad f= \frac{m}{100}

standard calculations

implied probability of payout

from betting odds we can determine the implied probability of that outcome occuring

\frac{\alpha}{\beta} \quad \Rightarrow \quad p=\frac{\alpha}{\alpha+\beta}

$p$ is the implied probability of odds $\frac{\alpha}{\beta}$

general winnings and stakes

when a bet wins the original stake is returned with the addition of the stake multiplied the odds

w=p+p \frac{\alpha}{\beta}=p\left(1+\frac{\alpha}{\beta}\right)

$w$ is the winnings (including the initial placement)

$p$ is the value of bet place

$\frac{\alpha}{\beta}$ are the odds

optimal ratios

specific winnings and stakes

each outcome has a winning associated with the stake placed on that outcome

w_1=p_1\left(1+\frac{\alpha}{\beta}\right) \quad w_2=p_2\left(1+\frac{\gamma}{\delta}\right) \quad \cdots \quad w_n=p_n\left(1+\frac{\psi}{\omega}\right)

$w_1,w_2,...,w_n$ are the corresponding winnings of every outcome

$p_1,p_2,...,p_n$ are the respective stakes placed on each outcome

introducing total stake, budget

bettor’s have limited budget for each event which is to be split amongst the outcomes in the optimal way

w_1=a_1 z\left(1+\frac{\alpha}{\beta}\right) \quad w_2=a_2 z\left(1+\frac{\gamma}{\delta}\right) \quad \dots \quad w_n=a_n z\left(1+\frac{\psi}{\omega}\right)

a_1+a_2+\cdots+a_n=1

$a_1,a_2,...,a_n$ are the ratios of the budget to be placed on each outcome (to be calculated)

$z$ is the total budget for the scenario (determined by operator)

condition for optimal ratio

the optimal ratio of placement of stakes occurs when the winnings of every outcome are equal

\max\left(a_1\left(1+\frac{\alpha}{\beta}\right)=a_2\left(1+\frac{\gamma}{\delta}\right)=\cdots=a_n\left(1+\frac{\psi}{\omega}\right)\left\{0<a_1+a_2+\cdots +a_n<1\right\}\right)

calculating optimal ratio

optimal ratio can be calculated by manipulation of the two expressions, making $a_n$ the subject

a_1\left(1+\frac{\alpha}{\beta}\right)=a_2\left(1+\frac{\gamma}{\delta}\right)=\cdots=a_n\left(1+\frac{\psi}{\omega}\right)

a_1+a_2+\cdots +a_n=1

for two outcome events

a_1=\frac{1+b_1}{\left(1+b_1\right)+\left(1+b_2\right)} \quad a_2=\frac{1+b_2}{\left(1+b_1\right)+\left(1+b_2\right)}

for three outcome events

a_1=\frac{\left(1+b_2\right)\left(1+b_3\right)}{\left(1+b_1\right)\left(1+b_2\right)+\left(1+b_1\right)\left(1+b_3\right)+\left(1+b_2\right)\left(1+b_3\right)}

a_2=\frac{\left(1+b_1\right)\left(1+b_3\right)}{\left(1+b_1\right)\left(1+b_2\right)+\left(1+b_1\right)\left(1+b_3\right)+\left(1+b_2\right)\left(1+b_3\right)}

a_3=\frac{\left(1+b_1\right)(1+b_2)}{\left(1+b_1\right)\left(1+b_2\right)+\left(1+b_1\right)\left(1+b_3\right)+\left(1+b_2\right)\left(1+b_3\right)}

for $n$ outcome events

a_{k \mathrm{th}}=\frac{\left(\frac{\prod_{j=1}^n\left(1+b_j\right)}{1+b_k}\right)}{\sum_{i=1}^n\left(\frac{\prod_{j=1}^n\left(1+b_j\right)}{\left(1+b_i\right)}\right)}

numerator equals ( product of all ( 1 plus every outcomes odds ) excluding the $i\text{th}$ odds )

denominator equals ( sum of all possible $n-1$ collection of ( 1 plus every outcomes odds ) )

quantity of items on denominator equals ( $n$ choose $n-1$ )