Recently I came across this article about sports betting arbitrage. The article gives formulas for calculating arbitrage profit and individual bet amounts for a two-outcome event. But it doesn't prove that those formulas will always yield the optimal arbitrage profit. Those formulas are reproduced below.
Let P(A) and P(B) be probabilities of the only two possible outcomes of an event. These probabilities are simply inverse of decimal odds. Let I be the total investment we are willing to make. Also let P(T) = P(A) + P(B). Then
Arbitrage Profit = [I / P(T)] - I
Amount to bet on outcome A = I * P(A) / P(T)
Amount to bet on outcome B = I * P(B) / P(T)
Two things I can't understand are:
- Why would the formulas above always yield the optimum profit for the given probabilities and investment?
- Are these formulas extendable to more than two outcomes?
I'm not a mathematician and trying to prove the above is doing my head in. Your help will be much appreciated!
Thanks in advance :)
Clarification - What is meant by 'Event'
Example of an event here would be a tennis match between Djokovic and Murray, the two outcomes being Djokovic and Murray. P(Djokovic) is obtained by taking reciprocal of decimal odds offered by the bookmaker for Djokovic. So if the odds for Djokovic win are 1.9 then P(Djokovic) = 0.526. In this case it is possible to have P(Djokovic) + P(Murray) = P(T) <> 1.0. When P(T) >= 1.0 there is no possibility of arbitrage. When P(T) < 1.0, then there is arbitrage profit. The above equations relate to the latter situation, i.e. when P(T) < 1.0.