Are the probabilities equal? I agree that they're both possible, but I don't have any clear intuition for why they should be the same -- or which would be larger if they're not the same.
The former deals with our expectations and the latter deals with probability. They are similar, but when we don't have a good grasp of the probability it might be better to phrase it in terms of our expectations.
>the two have the same upper bound on their probability?
Using only that information, yes.
Replace the names of the two events with "event A" and "event B" and it doesn't sound so weird.
The difference in those probabilities is the same. In each case, the event with the greater probability is 1-in-a-thousand times more likely to occur than the lesser.
Let's walk through an example with these numbers. Each case has 2 probabilities and we'll assign them each to the possible outcomes of a coin flip.
Case 1:
The probability of heads is 0 and the probability of tails is 0.001. We flip the coin 1,000 times. The ideal outcome is that we see 0 heads and 1 tails.
Case 2:
The probability of heads is 0.499 and the probability of tails is 0.500. We flip the coin 1,000 times. The ideal outcome is that we see 499 heads and 500 tails.
(You'll notice that the outcomes don't add up to 1,000 in either case, and that's because the probabilities don't add up to 1. That means there is actually a third possible outcome, but that can be ignored for the purpose of this example.)
They each differ by a result of 1 result per 1,000 trials. This is because probability is simply the likelihood of an event occurring. There is no super-linear pattern going on here. It is no harder to get from 0.7 to 0.9 probability than it is to get from 0.2 to 0.4. It is just a measure of likelihood.
What is causing people to misunderstand this? I wonder if you're thinking about probability distributions vs. probability itself?
They are not equivalent. The former term logically implies the latter term. That means the set of situations described by the former term is a subset of the situations described by the latter term. That means the probability of the former set should be less or equal than the probability of the latter set.
I think this is the correct way to phrase it. Just because the probabilities of each are both 50% doesn't mean it's more likely than not to get the same number of ones and zeros. It would just mean you're equally likely to get a few more ones as you are to get a few more zeros. But the counts are unlikely to be very far apart.
It is possible! Actually I remember? pointing out a similar concern in my probability class back in the day. The teacher's answer: that is precisely the difference between probablity and possibility :)
But you are assuming that each of those three possibilities has equal probability; can you explain the rationale for that? (it is clear why BB, BG, GB, and GG have equal probability in the unrestricted case, but less clear why BB, BG, and GB have equal probability in this restricted case)
Besides, this is just a rephrasing of the original article's argument, and doesn't counter mine at all. I am open to the possibility that there is a flaw in my argument, but where is it?
I am sorry but I am lost. If they both have probabilities or amplitudes of 1 would that not lead to a joint probability of 200% in the nonidentical case and 400% in the identical one?
I think you are making a big assumption with your number of "50%". It's true that there are two possible outcomes for either choice I could make. But what makes you say they are equally likely? To my mind, the estimating of that probability is actually the crux of the game.
Agreed 100% but the fact that these are different directly rebukes a common conception of what probability means. Many people believe that "what is the probability of X?" really means something like "if we were to re-create these exact conditions many times, what proportion of those iterations will X be true?"
This problem seems to be a good thought experiment demonstrating that those two questions are not equivalent. Which only makes one wonder what "probability" really means...
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