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:

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