Timeline for Three horse race
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 16, 2021 at 15:33 | comment | added | hexomino | Let us continue this discussion in chat. | |
| Dec 16, 2021 at 15:32 | comment | added | Stef | @hexomino Still, you are saying that there is a distribution of probabilities on the scenarios, even if you don't know the exact probabilities of each scenario. So you're answering a different problem where the scenarios are random events. In the given problem, the only random events are the finish-times of the horses. | |
| Dec 16, 2021 at 15:29 | comment | added | hexomino | @Stef Note that I'm not choosing a prior here. I'm simply stating that the prior is not one where Scenario 1 has probability 1. To refute that would be specifying the prior. | |
| Dec 16, 2021 at 15:23 | comment | added | Stef | @hexomino Because then you're answering a different problem. The problem statement describes a situation, and asks a question. Solving the problem means answering in all generality, so that the answer applies to any situation that fits the description. Placing a prior on the possible scenarios is really interesting and leads to an interesting answer, but it's an answer to a slightly different problem. Statisticians who choose their own priors are not solving a well-stated formal problem: they're modelling a problem themselves, then solving the problem under that model. | |
| Dec 16, 2021 at 15:22 | comment | added | hexomino | @Stef Are you familiar with the concept of a prior from Bayesian statistics? | |
| Dec 16, 2021 at 15:13 | comment | added | Stef | Oops, you're right, you're not making the assumption that the correlation is exactly zero. But what do you mean when you say "it's possible that the correlation is not 1"? Note that it is not correct to say "the probability that the correlation is not 1 is nonzero." Because there are no probabilities on this. The value of the correlation is not the result of a probabilistic experiment. The only probabilistic experiments in the problem are the horses finishing with a random time. | |
| Dec 16, 2021 at 15:06 | comment | added | hexomino | @Stef I'm not making the assumption that the correlation is exactly 0, that's wrong. I'm simply stating that it's possible that the correlation is not 1. | |
| Dec 16, 2021 at 14:19 | comment | added | Stef | @hexomino I completely agree that "since it is not mentioned that A and C are correlated then it is possible that they are not". Please note the difference between "we don't know anything about the correlation between A and C" and "We know that the correlation between A and C is exactly 0". Your answer makes the assumption that A and C are not independent and that their correlation is exactly 0. Loopywalt specifically does not make any assumption about A and C's correlation. In fact, they quite explicitly state that we have no information regarding that. | |
| Dec 16, 2021 at 12:19 | comment | added | Dmitry Kamenetsky | A and C are not correlated and that has never been mentioned. | |
| Dec 16, 2021 at 11:53 | comment | added | hexomino | @Stef So you're getting into loopywalt's argument which is quite a philisophical one. My argument to that would be since it is not mentioned that A and C are correlated then it is possible that they are not. | |
| Dec 16, 2021 at 11:40 | comment | added | Stef | Yes, if A is 60 and C is 57, then B wins. But from the information given in the problem, you cannot conclude that the joint probability of A=60, C=57 is nonzero. Similarly, from the information given in the problem, you cannot conclude that the joint probability of C=53, A=60 is nonzero. | |
| Dec 16, 2021 at 11:36 | comment | added | hexomino | @Stef No, it's not an assumption, it's very clear from the information given - B wins if A is 60 and C is 57, C wins if C is 53 and A is 60. | |
| Dec 16, 2021 at 11:32 | comment | added | Stef | But how do you know that it's possible? That's not stated in the problem. You're making an assumption there. | |
| Dec 16, 2021 at 11:28 | comment | added | hexomino | @Stef They are both non-zero because there are scenarios where they both can win. If something is possible, then it has probability greater than zero. | |
| Dec 16, 2021 at 11:21 | comment | added | Stef | Where did you get "and since they are both non-zero" from? | |
| Dec 16, 2021 at 11:14 | comment | added | hexomino | @Stef Because the sum of the probabilities will be 1, the sum of the probabilities of B or C winning is <= 1/2 and since they are both non-zero, they must also be both less than 1/2 | |
| Dec 16, 2021 at 11:11 | comment | added | Stef | I don't understand this argument: "Given that there are scenarios where Horse B or C could win, this means that neither of their chances can be as high as 1/2". Why could neither of their chances be as high as 1/2? | |
| Dec 15, 2021 at 2:25 | vote | accept | Dmitry Kamenetsky | ||
| Dec 14, 2021 at 14:16 | history | edited | Dmitry Kamenetsky | CC BY-SA 4.0 |
deleted 1 character in body
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| Dec 14, 2021 at 12:45 | history | answered | hexomino | CC BY-SA 4.0 |