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If you compute the 95% CI for a population mean from a given sample, apparently it is taboo to say something like "there is a probability of 0.95 that the true mean lies in this specific CI".

Yet when you look at what "confidence" means, it is a statement about all the possible CI (computed using the same sample size), which says that 95% of these CI will contain the true mean.

When we know that if 1% of the tickets in a lottery have the predetermined winning number, we have no problem saying (after we draw a ticket) the probability we drew a winning ticket is 1%.

Why do we have such a problem making a similar probability statement for a given CI?

Matt Brenneman
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  • I think you will find the information you need in the linked thread. Please read it. If it isn't what you want / you still have a question afterwards, come back here & edit your question to state what you learned & what you still need to know. Then we can provide the information you need without just duplicating material elsewhere that already didn't help you. – gung - Reinstate Monica Dec 07 '16 at 18:44
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    Note that "*after* we draw a ticket", the probability we drew a winning ticket is most certainly *not* 1%. It is either 0% or 100%, although we won't know which until we look at the ticket. – gung - Reinstate Monica Dec 07 '16 at 18:46
  • The lottery example is not quite right because the drawing has not happened. We can't pick up an expired losing ticket to make the same winning claim. Sampling is more like the latter, the samples have been drawn. – Penguin_Knight Dec 07 '16 at 18:47
  • @Penguin_Knight I fail to see the analogy. The true result is *never* known in a CI just like the true result is not known before the drawing. We have the exact same state of ignorance in both cases – Matt Brenneman Dec 07 '16 at 18:51
  • Your state of ignorance is irrelevant. The number on the ticket is what it is whether you have looked at it or not. It is either the winning number or it isn't the winning number whether you know what the number is or don't know. It may help you to read my answer at the linked thread. – gung - Reinstate Monica Dec 07 '16 at 19:02
  • @gung It seems to me it is a probability I won is one percent even after I draw it. If I had to compute the expected value of my ticket for someone who wanted to buy it, how would I do it? E(X) = 0.01*Payoff + 0.99*(whatever I paid for ticket). Isn't that 0.01 the probability that my ticket is a winning ticket? – Matt Brenneman Dec 07 '16 at 19:05
  • That doesn't apply. The [expected value](https://en.wikipedia.org/wiki/Expected_value) is the long-run weighted average of a random variable, where the weights are the probabilities / densities of the values the RV could take. When you have a *drawn* ticket, you have a constant. Until you turn the ticket over, it is an [unknown constant](https://en.wikipedia.org/wiki/Constant_(mathematics)), but it is not a [random variable](https://en.wikipedia.org/wiki/Random_variable). – gung - Reinstate Monica Dec 07 '16 at 19:24
  • You could call it a degenerate random variable & say it's expected value is \$0, or the prize value, but that you don't know which. From an economic perspective, you could try to price it rationally as a function of the prize value times your Bayesian prior probability that it is a winning ticket. Moreover the perspective buyer can do likewise. (Since your priors may not match, there may not be a sale, or one of you may get what they consider to be a good deal.) – gung - Reinstate Monica Dec 07 '16 at 19:26
  • @gung Thank you for your clear explanations. I'd like to understand logic and epistemology at the level you do. I hope I don't seem intrusive and I know this is off topic but I would very much like to learn what subjects you studied and where. – Matt Brenneman Dec 08 '16 at 22:38
  • I'm not sure I follow you. There's nothing wrong w/ your questions, so there's no need to be apologetic. Your thoughts on this are very intuitive; it's typically what people think. You can get a fuller sense of what I'm trying to explain in my answer at the linked thread. – gung - Reinstate Monica Dec 08 '16 at 23:15
  • @gung Thank you. I am not sure if this makes sense, but I would like to be able to understand these matters more clearly and learn to think from first principles like you do. I know the answer because you told me, but I would like to learn how to dissect and analyze such arguments on my own and moreover understand the relevant rules of logic that apply. – Matt Brenneman Dec 09 '16 at 00:04
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    That's very generous. I'm not sure what I can tell you, though. Spending time reading around on CV certainly exposes you to a lot of stuff, & that really helps. Much of the stuff I know now I'm not really sure where I learned. – gung - Reinstate Monica Dec 09 '16 at 00:31

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