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$\quad$ I have a friend of mine who is a bit of a gambler ask me this question. He is of poor mathematical background, but has a sense of logic and will probably accept a logical answer in the natural language.

$\quad$ So my friend asked me if the probability of an outcome would change in consecutive trials of playing roulette. I said that it will not, and he needed no further verification. He only asked if it is because the roulette has no memory. I said he could put it that way if he liked.

$\quad$ Then the person asked me if the same is true for soccer matches. I told him that, despite the influence on soccer by other factors, the same should hold. My friend insisted on the contrary, and I didn't know what to say to him.

$\quad$ I would like to be able to explain to him why soccer matches are independent events. If this is not true, what is the dependence? Would considering a particular team change anything. Furthermore, what restrictions can we apply to ensure independence? Having the same opponent team?

p.s. I wouldn't mind an answer including mathematics in addition to any other answers.

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closed as off topic by Qiaochu Yuan Feb 27 '13 at 20:01

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If there independent events then by definition your friend is wrong, but say the past wins or losses effected the self esteem of the team and in turn there performance, they wouldn't be truly independent events, how can you really take into account all these things, but strictly speaking if I had to chose one or the other I would say your friend is wrong. –  Ethan Feb 27 '13 at 19:35
    
I'm not really sure what the sample space is here. –  Rahul Feb 27 '13 at 19:42
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Surely the matches of a single team depend on the previous matches. But the effect is largely psychological: a team confident in its ability will play better than it would after a couple of setbacks. At some point the reverse factors will come to the fore - a team becomes overconfident after a long winning streak or after a slump really wants to prove itself. Surely somebody has collected huge volumes of data on this, and has modelled the statistics of winning/losing streaks. Major League Baseball in the US and the Premier League in the UK come to mind... –  Jyrki Lahtonen Feb 27 '13 at 19:43
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This is not a mathematical question. To answer it, one should define the set of games one is interested in and perform some statistical test to check whether independence holds or not. –  Did Feb 27 '13 at 19:44
    
Agree with Did. This is an empirical question about soccer matches. –  Qiaochu Yuan Feb 27 '13 at 20:01
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1 Answer

up vote 1 down vote accepted

Two events $B$ and $R$ are independent if and only if $P(B)=P(B|R)$. This is certainly the case if $B$ is black ball, $R$ is ref ball in roulette. However, it is not possible to conclude such a result in football games. Of course, the performance of the players in the last game affect their performance in the next game.

To my knowledge, while calculating the odds bookies take into account the following criteria, the recent performance of the teams, the matches they played between themselves, the place they play (away or home) and so on.. On the other hand, the odds in roulette is fixed. I hope this comparison helps you in explaining the situation to your friend.

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