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I'm trying to use odds data from bookmakers to estimate the expected number of goals in a game. We have these known facts:

  • P(o4.5) = 0.573
  • P(o5.5) = 0.458
  • P(o6.5) = 0.279

P(o4.5) is the probability that there will be more than 4.5, P(o5.5) more than 5.5 and so on. The probability that there will be less than 4.5 is the inverse: 1 - P(o.45).

Can you from these facts estimate the expected number of goals in the game? Something like

P(oX) = 0.5 

What is the value of the X variable? Is the information given above enough to figure it out? Is it enough if you make some assumptions like on the shape of the goal distribution and so on?

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    $P(\text{Goals} \ge 4.5) = 0.353 < 0.458 = P(\text{Goals} \ge 5.5)$ doesn't make sense IMHO ... if there are at least 5.5, there are at least 4.5 also ...2012-03-22
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    How can the probability that there be more than 4.5 be lower than the probability that there be more than 5.5? And what is a half goal anyway? What kind of game is this?2012-03-22
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    Sorry I suck. I have updated the numbers. Half goal is a convenient notation used by bookmakers. P(oX.5) means (x+1) or more goals. P(uX.5) means X or less goals.2012-03-22
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    @Raskolnikov the half goal is done to exclude confusion. If I say "more than 4.5" it means 5, 6, 7,... goal. If I say "more than 4" maybe i can have doubt if the 4 is included.2012-03-22
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    @Kolmo: I see, but wouldn't it be easier to use $>$ instead of $\geq$ to avoid such confusion. Anyway, I guess every field has its own conventions.2012-03-22

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