# Betting Strategies – Part 1 – Maximizing Expected Return

This post is the first of two to discuss strategies for sporting events where there are three or more possible outcomes. Examples include football, horse racing, the first try scorer in rugby, and highest run scorer in cricket. For all of these bets you could pick more than one outcome and make a profit if one of your picks is correct. For example you could bet on three different cricketers to score the highest runs, and set your bets so that if any of your selections wins, your selection’s profit more than cancels out your losses from your other selections.

Two betting strategies will be discussed in this series: ‘Maximize expected return’, and ‘Maximize the return to risk ratio.’ The former will be discussed here, and the latter will be discussed in my next post. Most of the content has been sourced from chapter 20 in the book Paul Wilmott introduces quantitative finance, by Paul Wilmott.

### Maximize expected return

This strategy is the simplest of the two, and involves betting on the outcome that provides the greatest expected profit. Unlike the second strategy in this series, this strategy pays no attention to risk.

Let’s look at the ‘high bat’ for Australia’s first innings in the second Ashes test in 2009. Consider each outcome in turn and determine your own probability that each option will win. Then run the calculation (ODDS x PROBABILITY) – 1 to determine your expected return for each pick.

 Batsman Bookmaker odds Your probability Expected return Ricky Ponting 4.00 20% -20% Simon Katich 5.00 23% 15% Phillip Hughes 5.25 15% -21% Michael Clarke 5.50 22% 21% Michael Hussey 8.00 8% -52 Marcus North 7.50 6% -55% Brad Haddin 10.00 5% -50% Mitchell Johnson 31.00 1% -69% Nathan Hauritz 81.00 0% -100% Peter Siddle 201.00 0% -100% Ben Hilfenhaus 251.00 0% -100% Total probability 100%

Based on these probabilities you would select Michael Clarke because he provides the highest expected return. Your expected return would be 21% with a standard deviation of 228% (the methodology of calculating standard deviation is discussed in the next post).

Note that if none of the expected returns were positive, your best strategy would be to not make any bet.

This strategy doesn’t take risk into account. In my next post I will outline the ‘maximize the return to risk ratio’ strategy. This will moderate your exposure to risk and it is a mathematically superior strategy.