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You are committing the gambler’s fallacy. There’s no reason to expect your “luck” (whatever you mean by that) to decrease rather than increase the next time you

You are committing the gambler’s fallacy. There’s no reason to expect your “luck” (whatever you mean by that) to decrease rather than increase the next time you race. This is different from saying that you are likely to win again. You’re just as likely to win as you were the first time around, which may be high or low depending on ability.

Jtsummers 8y ago
Drop out the ability part. Let's focus on a pure luck scenario: Take 2 dice (2d6) and roll them. You get a 12. What is more probable on your next roll? Another 12 or a 7? With regard to the race, if you do really well, much better than your average, without any fundamental change to your ability that's the same as rolling a 12. It's an unlikely (though possible) event that happened to occur. Over a series of races, however, we would not anticipate a repeat of that performance (that is, it's a low probability event, and a series of such low probability events has even lower probability). The gambler's fallacy goes the other way. It's the belief that if we've had a string of bad luck [0], then we should wager on good luck being around the corner. That is, the gambler assumes a low probability event is "inevitable" after a series of high probability events. Regression to the mean: A high probability event should be anticipated after a low probability event. [0] In gambling this usually means getting close to the average, that is: the average hand is the lowest valued hand, the rarest hand is the highest valued. So a gambler making this fallacy would wager on seeing a royal flush after several hundred hands of only seeing pairs.
- antidesitter 8y ago
  I understand what the GP was saying now (see my sibling comment). The gambler’s fallacy can apply the other way, e.g. “This die has rolled over 3 many times, so it will probably roll 3 or lower next time.” So that's not really the issue. Your dice example is not illustrating the point the GP is trying to make. It’s just comparing the probabilities of two outcomes of the same distribution (the sum-of-two-dice distribution). This has nothing to do with the selection effect the GP is talking about. That would be recording the value of the die that rolled highest (i.e. taking the maximum over dice) and then focusing on the same die next time (i.e. not taking the maximum over dice). It’s comparing the probability distribution of max {X_1, X_2, X_3, ...} with the probability distribution of any individual X_i. If A is sampled from the former and B is sampled from the latter, P(B < A) > P(B > A).
btilly 8y ago
No, I am not committing the gambler's fallacy. Though I can see why you might think that. There are enough things that sound similar. I'm instead talking about a more subtle detail. Which is that selecting a group based on performance results in selecting people in part for having been lucky. They were lucky to have done that well that time, were lucky to be in your group, and their future performance probably won't be as good. The result is called regression to the mean. See https://academic.oup.com/ije/article/34/1/215/638499 https://academic.oup.com/ije/article/34/1/215/638499 for an example of statisticians talking about it. See http://onlinestatbook.com/stat_sim/reg_to_mean/index.html http://onlinestatbook.com/stat_sim/reg_to_mean/index.html for a more introductory tutorial. And see http://wmbriggs.com/post/63/ http://wmbriggs.com/post/63/ for an example of an article discussing this counterintuitive phenomena in the context of sports. (Namely the "Sports Illustrated curse" - the future performance of athletes whose performance was good enough to get them on the cover of Sports Illustrated drops after their article appears.)
- antidesitter 8y ago
  I think I see what you mean now. If you roll 20 dice and select the highest, the next time you throw that very same die it’ll probably have a lower value, simply as a consequence of outcome-dependent selection + a fixed probability. It’s not that the probability has changed, just that you didn’t choose the first outcome according to the true generating distribution. Comparing this highest value with the highest value across all coins when you toss them all again a second time is a different matter. It’s no more likely to be lower than higher. It's best to avoid using the term “luck” altogether in discussions of probability, since it’s easy to misinterpret.
  - btilly 8y ago
    That's exactly the right idea.

Jtsummers 8y ago

Drop out the ability part. Let's focus on a pure luck scenario: Take 2 dice (2d6) and roll them. You get a 12. What is more probable on your next roll? Another 12 or a 7? With regard to the race, if you do really well, much better than your average, without any fundamental change to your ability that's the same as rolling a 12. It's an unlikely (though possible) event that happened to occur. Over a series of races, however, we would not anticipate a repeat of that performance (that is, it's a low probability event, and a series of such low probability events has even lower probability). The gambler's fallacy goes the other way. It's the belief that if we've had a string of bad luck [0], then we should wager on good luck being around the corner. That is, the gambler assumes a low probability event is "inevitable" after a series of high probability events. Regression to the mean: A high probability event should be anticipated after a low probability event. [0] In gambling this usually means getting close to the average, that is: the average hand is the lowest valued hand, the rarest hand is the highest valued. So a gambler making this fallacy would wager on seeing a royal flush after several hundred hands of only seeing pairs.

antidesitter 8y ago
I understand what the GP was saying now (see my sibling comment). The gambler’s fallacy can apply the other way, e.g. “This die has rolled over 3 many times, so it will probably roll 3 or lower next time.” So that's not really the issue. Your dice example is not illustrating the point the GP is trying to make. It’s just comparing the probabilities of two outcomes of the same distribution (the sum-of-two-dice distribution). This has nothing to do with the selection effect the GP is talking about. That would be recording the value of the die that rolled highest (i.e. taking the maximum over dice) and then focusing on the same die next time (i.e. not taking the maximum over dice). It’s comparing the probability distribution of max {X_1, X_2, X_3, ...} with the probability distribution of any individual X_i. If A is sampled from the former and B is sampled from the latter, P(B < A) > P(B > A).

btilly 8y ago

No, I am not committing the gambler's fallacy. Though I can see why you might think that. There are enough things that sound similar. I'm instead talking about a more subtle detail. Which is that selecting a group based on performance results in selecting people in part for having been lucky. They were lucky to have done that well that time, were lucky to be in your group, and their future performance probably won't be as good. The result is called regression to the mean. See https://academic.oup.com/ije/article/34/1/215/638499 https://academic.oup.com/ije/article/34/1/215/638499 for an example of statisticians talking about it. See http://onlinestatbook.com/stat_sim/reg_to_mean/index.html http://onlinestatbook.com/stat_sim/reg_to_mean/index.html for a more introductory tutorial. And see http://wmbriggs.com/post/63/ http://wmbriggs.com/post/63/ for an example of an article discussing this counterintuitive phenomena in the context of sports. (Namely the "Sports Illustrated curse" - the future performance of athletes whose performance was good enough to get them on the cover of Sports Illustrated drops after their article appears.)

antidesitter 8y ago
I think I see what you mean now. If you roll 20 dice and select the highest, the next time you throw that very same die it’ll probably have a lower value, simply as a consequence of outcome-dependent selection + a fixed probability. It’s not that the probability has changed, just that you didn’t choose the first outcome according to the true generating distribution. Comparing this highest value with the highest value across all coins when you toss them all again a second time is a different matter. It’s no more likely to be lower than higher. It's best to avoid using the term “luck” altogether in discussions of probability, since it’s easy to misinterpret.
- btilly 8y ago
  That's exactly the right idea.