I was considering an example with 10^100 scientists. I thought that since there would be a lot more scientists who got 25 big in the 1⁄4 scenario than in the 3⁄4 scenario (about 9.18 10^98 vs. 1.279 10^75), you’d be more likely to be first the 3⁄4 scenario. But this forgets about the probability of getting an improbable result.
In general, if there are N scientists, and the probability of getting some result is p, then we can expect Np scientists to get that result on average. If the order is shuffled as you suggest, then the probability of being the first to get that result is p * 1/(Np) = 1/N. So the probability of being the first to get the result is the same, regardless of the likelihood of the result (assuming someone will get the result).
EDIT: It occurs to me that I might have been thinking about the probability of being selected by Al conditional on getting 25⁄100. In that case, you’re a lot more likely to be selected if the pond is 3⁄4 big than if it is 1⁄4 big, since WAY more people got similar results in the latter case. JGMWeissman was probably thinking the same.
You’re right, thanks.
I was considering an example with 10^100 scientists. I thought that since there would be a lot more scientists who got 25 big in the 1⁄4 scenario than in the 3⁄4 scenario (about 9.18 10^98 vs. 1.279 10^75), you’d be more likely to be first the 3⁄4 scenario. But this forgets about the probability of getting an improbable result.
In general, if there are N scientists, and the probability of getting some result is p, then we can expect Np scientists to get that result on average. If the order is shuffled as you suggest, then the probability of being the first to get that result is p * 1/(Np) = 1/N. So the probability of being the first to get the result is the same, regardless of the likelihood of the result (assuming someone will get the result).
EDIT: It occurs to me that I might have been thinking about the probability of being selected by Al conditional on getting 25⁄100. In that case, you’re a lot more likely to be selected if the pond is 3⁄4 big than if it is 1⁄4 big, since WAY more people got similar results in the latter case. JGMWeissman was probably thinking the same.