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I am going through Trevor Hastie's Classification Techniques.

Its says Odds are traditionally used instead of probabilities in horse-racing.

I still don't understand how they relate more naturally to the correct betting strategy?

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It's not that they relate more naturally to the correct betting strategy per se, it's that they're much more easily interpretable while on the horse track.

Consider the following example: the probability of horse A to win is 66% percent. In conventional odds notation, this is represented as a 2:1 bet, which is very easily interpretable as a £2 win for each £1 bet, if the horse does win. For the layman, the 66% probability does not lend itself so easily to interpretation in terms of potential winnings.

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In a logistic regression model, odds ratio provide a more coherent solution as compared to probabilities. Odds ratio represent the constant effect of an independent variable on a dependent variable. Here, being constant means that this ratio does not change with a change in the independent (predictor) variable. Odds ratio in this sense provide a much easier way of comprehending the marginal effects.

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