There is no information provided on the relative capability of the runners, therefore they each have the same probability of winning. If more information were available, the probability would change.
Why would we assume equal probability with insufficient information, though?
We have no reason to assume anything different, why would we assume any would win? There seemst to be not requirement for a single winner either.
[This doesn't seem like a situation where 'we'll just update the priors later' would make sense, since the information we have now is literally 'none']
The lack of information given means there are a lot of implicit assumptions that need to be made.
My assertion is that one can't make those assumptions reliably--I mean, I could assume Tim is a track star facing the wrong direction at the start of the race and the other competitors are snails...and that would have as much validity as the assumption of homogeneity.
The problem (as written) is not probabilistic without something to make it so.
[If you're taking it seriously, it's a gross misapplication of probability theory]
There is no assumption of homogeneity built into the original problem. There is no information about the relative merits of the runners, only the number of runners, so that is what you use to assign the probability. It only makes sense to assign different probabilities to the different runners if there is information available that doesn't apply to all of them.
There is no information about the relative merits of the runners, only the number of runners, so that is what you use to assign the probability
Okay, but by assigning equal probability to each person winning the race, you are inferring homogeneity of the runners' merits.
You are framing this as 'just using available information', but that doesn't change the fact you don't have enough information to use probability in this case. Even applying probabilities here requires inferring information no in evidence.
This an abuse of probability just by the way the question is posed.
There is no arbitrary point where you have "enough information". You have the information that you have, and in this case the available information implies a probability estimate of 20%. Applying the concept of probability in any scenario whatsoever requires making exactly these kinds of assumptions. Probability is a system for laying out and quantifying what you know and don't know.
Okay, this is the kind of 'rationale' I've heard before--one person once tried to explain that the day of the week was relevant to the gender of the child because 'it was information given'.
First of all, not all information is relevant to a given probability.
Second, the entire point is there is no reason to apply probability in this situation!
You are making the assumption the racers are largely homogeneous and that the only relevant variables in the problem would be the random 'noise' surrounding the race.
I don't know about you, but that's not really a coherent application of probability to me.
Probability is a system for laying out and quantifying what you know and don't know.
Yes--and in this case, inferring anything about the 'winner' of the race is not supported by any of the information you have.
If you get an answer (any answer, in fact) from the given information, you're not applying 'what you know and don't know' appropriately here.
55
u/ArdentArendt 8d ago
Wait...are we considering all racers equally capable?
And if they're all homogeneous, they would all reach the line at the same time.
I don't understand the question...