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An English soccer team plays a series of matches again different opponents, of varying ability. A bookmaker offers odds for each match as to whether it will be a home win, away win, or draw. Part-way through the season, the team has played n matches, and has drawn k of them, which is more than might be expected from the odds. What is the probability that the bookmaker is mis-pricing the odds on these matches, rather than just being unlucky? If the bookmaker continues to price the team's remaining matches in a similar way, and I bet $1 that each one will be a draw, what is my expected return?

As an added bonus I will email the name of the team to the author of the best answer :-)

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There is a non-zero probability that the author of this question is a bookmaker looking for some action :-) BTW, don't you think the bookmaker has the same information you have, and can adjust his rates accordingly? Also, you should define better n, k, what you expected from the odds and why. – user9346 Apr 9 '11 at 9:02
On the contrary, I'm trying to find a good bet to place with the bookmaker :-) In the most striking case that I've identified, n is 38 and k is 15. The odds of course vary from match to match, but given the odds for a particular match, you can calculate the implied probability that the bookmaker thinks there is of each outcome. And if they consistently get it wrong, you can obtain a reasonably reliable return. – user9341 Apr 9 '11 at 9:37
Predicting sports results is a difficult topic - as you can see it is studied by scientists… I think that the approach taking into account only number of wins/draws/loses is overly simplistic. However, from my real life experience, I have the feeling that the bookies react not only to probabilities of expect outcomes but also to the behaviour of bettors. Typically, often the odds on an outsider tend to be higher than what you would expect based on probabilities of outcomes. – Martin Sleziak Apr 9 '11 at 10:00
Just wondering aloud: shouldn't this question be better suited for stats.stackexchange, instead of here? – Willie Wong Apr 9 '11 at 10:40
There are so many issues here it's hard to know where to get started. (1) A rational bookmaker is not trying to set the odds to the actual probability of occurrence of each event; they're trying to construct an arbitrage. (2) However, empirical studies show that with lots of bettors and money bet, the odds do largely reflect actual probabilities of occurrence. (3) There is a (huge!) difference between evaluating whether a particular team has mispriced odds and whether some team among many has mispriced odds. The process of searching moves you from the former situation to the latter. – cardinal Apr 9 '11 at 14:29

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