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I am currently studying logistic regression. So the Probability(p(x)) of assigning an outcome is calculated as :

$\frac{p}{1-p} = e^{\beta_0 +\beta_1X} $

Further, I read that the fraction on the left is also called odds, frequently used in betting references.

So I went to a betting website, here were the odds given by them for a football match. Chelsea vs West Brom ( Win Draw Win : 1.5, 4, 7.5)

Now I understand that they would have used a program to come up the odds.

But I figured that using the odds, I should be able to calculate the probabilities, that their software is assigning to each of the outcome.

When doing the calculations : I got 0.4 prob of Chelseas win, 0.2 of Draw and 0.11 chance of West Brom Win.

Why is this not adding up to 1?

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    $\begingroup$ Are these what they think the odds are or do they specify what the payoffs are? Ordinarily, bookies could care less about estimating odds, so I would be surprised if these had much to do with actual probabilities. $\endgroup$ – whuber Jan 13 '16 at 18:25
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Decimal odds can be turned into their implied probabilities via the formula: $P=1/odds$. If you do this and sum you get 1.05 so I am guessing there is a small rounding errors in the decimal odds reported.

In response to the comment below, yes I am sure since these are,decimal odds . What is missing in this formula is the overhead that a "bookie" often uses to make the odds lower.

This formula is: $odds = 1/(P + \sigma)$, If you assume that the $\sigma$ are the same you can calculate this bias, and the probability will sum to 1.

Also see this post: How to convert sport odds into percentage? Asking essentially the same question.

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  • $\begingroup$ Also, this is not why the probabilities do not add up. $\endgroup$ – Cliff AB Jan 13 '16 at 18:45
  • $\begingroup$ @cliffab do tell $\endgroup$ – Repmat Jan 13 '16 at 18:49
  • $\begingroup$ hmm, I just learned about decimal odds. $\endgroup$ – Cliff AB Jan 13 '16 at 19:05
  • $\begingroup$ (I deleted my earlier comment because I didn't realise you were referring to payouts rather than true odds. I hadn't heard of the term "decimal odds" before.) $\endgroup$ – mark999 Jan 13 '16 at 20:00
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You are both right. It all depends on how you express the odds. This time I think it is a decimal representation so it means: how much you get for 1$.

In this case you will will $1.5\$$ for every $1\$$ you pay. Then the formula is $$p = \frac{1}{odds}$$

If you have fractional odds like $3:2$ then the formula is $$p = \frac{odds}{1+odds}$$.

Have a look for example here

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