Timeline for Three horse race
Current License: CC BY-SA 4.0
13 events
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| Dec 28, 2021 at 4:29 | comment | added | justhalf | @Stef, btw, I found more insightful observations and discussions on my answer. Do you have a comment (if you want)? Especially since I think I've nailed down the differences in assumption in the two views, but I'm not sure how to resolve them (if even possible) in some way. | |
| Dec 28, 2021 at 4:27 | comment | added | justhalf | @JasonGoemaat "and with the given information it should be assumed to be completely random." Agree! But note that "it should be assumed to be completely random" is not equal to "it should be assumed to be not dependent". They may still be dependent, or independent, we don't know. | |
| Dec 17, 2021 at 8:06 | comment | added | Stef | Yes, you're absolutely right. Horse C's result is completely random, with 50% chance of finishing in 53 seconds and 50% chance of finishing in 57 seconds. And we have absolutely no information about how correlated this is with Horse A's finishing times. And since we have no information, we can't just assume. Perhaps Horse C wins 25% of the time, or perhaps Horse C wins 0% of the time, or perhaps 50% of the time, or perhaps some other number. There is no way to know. | |
| Dec 17, 2021 at 7:23 | comment | added | Jason Goemaat |
@Stef I disagree. Given the information in the problem, Horse C will finish the race in 53 or 57 seconds with both events being equally likely. It does not say that horse C will finish in 53 seconds if fed broccoli and 57 seconds if fed lasagna. The statement does not declare any dependent factors that would result in the discrepancy, and with the given information it should be assumed to be completely random.
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| Dec 16, 2021 at 14:47 | comment | added | Stef | "I think it's contrived to assume some dependency that has to itself be an equal probability for both statements to be true" I agree; it's more than just contrived, it's wrong. We can't assume anything that isn't spelled out in the problem statement. We cannot assume that the events are not independent, and we cannot assume that the events are independent. We have to solve the problem with only the information given in the problem statement. And the answer is: P(A wins) = 50%, and P(B wins or C wins) = 50%, but P(C wins) and P(B wins) cannot be calculated because we lack information. | |
| Dec 16, 2021 at 14:41 | comment | added | Jason Goemaat | But this is just a puzzle and in a real world scenario there would never be a horse that would run a race in two different times with equal probability in the first place, no matter how you tune the factors. | |
| Dec 16, 2021 at 14:40 | comment | added | Jason Goemaat | In your example then Horse A would win 1/2 the time (when all horses are fed broccoli) and Horse B would win the other 1/2 the time (when all horses are fed lasanga), right? I would expect such dependencies to be spelled out in the question. I think it's contrived to assume some dependency that has to itself be an equal probability for both statements to be true, however I can see the value in asking "why" Horses A and C have such different timings with equal probability. In a real world scenario I would think track conditions would be a more likely factor. | |
| Dec 16, 2021 at 14:14 | comment | added | Stef | However, in the problem statement, we are not told whether the horses are fed the same menu, or a different menu, or independent menus, or something else. Any assumption we make in that regard is an extra assumption. | |
| Dec 16, 2021 at 14:13 | comment | added | Stef | Then you can safely conclude: "Horse A will finish the race in 50 or 60 seconds with both events being equally likely." and "Horse C will finish the race in 53 or 57 seconds with both events being equally likely." Yet the events "A finishes in 50 seconds" and "C finishes in 53 seconds" are certainly not independent. | |
| Dec 16, 2021 at 14:11 | comment | added | Stef | No, it doesn't. Imagine the following situation: everyday the horses are fed the same menu. When horse A is fed broccoli, it runs in 50 seconds. When horse C is fed broccoli, it runs in 53 seconds. When horse A is fed lasagna, it runs in 60 seconds. When horse C is fed lasagna, it runs in 57 seconds. Now, I tell you that the probability that today's menu is broccoli is 50%, and the probability that today's menu is lasagna is 50%. Then you can safely conclude: | |
| Dec 16, 2021 at 14:05 | comment | added | Jason Goemaat | When two times are given with 'equally likely' specified, doesn't that mean there are no dependencies? | |
| Dec 16, 2021 at 11:19 | comment | added | Stef | This answer makes the extra assumption that the events listed in the problem are independent, which wasn't stated in the question; yet the word "independent" doesn't even appear in this answer. So, you're basing your calculations on an unspoken extra assumption. | |
| Dec 14, 2021 at 21:48 | history | answered | Jason Goemaat | CC BY-SA 4.0 |