# Betting Maths

For an event $$A$$ with decimal Back Odds of $$B$$ and Lay Odds of $$L$$, where $$P(A) = p$$ and $$P(\overline{A})=q$$.

## Kelly ratio for Back Odds

\begin{align} f_{b} = \frac{pB - 1}{B - 1} \end{align}

## Kelly ratio for Lay Odds

\begin{align} f_{l} = 1 - Lp \end{align}

## Decimal odds to net fractional odds

e.g. betting $10, on win, rewards$4 plus wager; then $$b = 0.4$$ and $$l = 2.5$$

\begin{align} b & = B - 1 \\ l & = \frac{1}{L - 1} = \frac{1}{b} \end{align}

## Hedging a bet

Where $$W$$ is stake + payout of existing bet (e.g. betting $10 on win with$4 wager; then $$W = 14$$).

To hedge a bet with Lay Odds of $$L$$, the backers stake, $$X$$ is:

\begin{align} X = \frac{W}{L} \end{align}