“We care about these odds because the laws of probability tell us to use them”—I’m not disputing your calculation, just explaining what you’ve actually calculated and why it isn’t relevant.
“Pre-committed at the start to guess if and only if...”—Given a scenario, we can calculate the probabilities when particular events occur. For example, if we have a deck of cards and we reveal cards one by one, we can ask about the probability that the next card is a king given that the previous card was a king. One way to describe this scenario would be to ask about the probability of guessing the correct card if you promise to guess the next card whenever you see a king and to not guess whenever you don’t. If you break this promise, then it may alter the chance of you guessing correctly. Is my use of language clear now?
You are simply assuming that what I’ve calculated is irrelevant. But the only way to know absolutely for sure whether it is irrelevant is to actually do the calculation! That is, if you have information X and Y, and you think Y is irrelevant to proposition A, the only way you can justify leaving out Y is if Pr(A | X and Y) = Pr(A | X). We often make informal arguments as to why this is so, but an actual calculation showing that, in fact, Pr(A | X and Y) != Pr(A | X) always trumps an informal argument that they should be equal.
Your “probability of guessing the correct card” presupposes some decision rule for choosing a particular card to guess. Given a particular decision rule, we could compute this probability, but it is something entirely different from “the probability that the card is a king”. If I assume that’s just bad wording, and that you’re actually talking about the frequency of heads when some condition occurs, well now you’re doing frequentist probabilities, and we were talking about *epistemic* probabilities.
I’m not using the word irrelevant in the sense of “Doesn’t affect the probability calculation”, I’m using it in the sense of, “Doesn’t correspond to something that we care about”.
Yeah, I could have made my language clearer in my second paragraph. I was talking about the “probability of guessing the correct card” for a particular guessing strategy. And the probability of the next card being a king over some set of situations corresponds to the probability that the strategy of always guessing “King” for the next card gives the correct solution.
Anyway, my point was that you can manipulate your probability of being correct by changing which situations are included inside this calculation.
“We care about these odds because the laws of probability tell us to use them”—I’m not disputing your calculation, just explaining what you’ve actually calculated and why it isn’t relevant.
“Pre-committed at the start to guess if and only if...”—Given a scenario, we can calculate the probabilities when particular events occur. For example, if we have a deck of cards and we reveal cards one by one, we can ask about the probability that the next card is a king given that the previous card was a king. One way to describe this scenario would be to ask about the probability of guessing the correct card if you promise to guess the next card whenever you see a king and to not guess whenever you don’t. If you break this promise, then it may alter the chance of you guessing correctly. Is my use of language clear now?
You are simply assuming that what I’ve calculated is irrelevant. But the only way to know absolutely for sure whether it is irrelevant is to actually do the calculation! That is, if you have information X and Y, and you think Y is irrelevant to proposition A, the only way you can justify leaving out Y is if Pr(A | X and Y) = Pr(A | X). We often make informal arguments as to why this is so, but an actual calculation showing that, in fact, Pr(A | X and Y) != Pr(A | X) always trumps an informal argument that they should be equal.
Your “probability of guessing the correct card” presupposes some decision rule for choosing a particular card to guess. Given a particular decision rule, we could compute this probability, but it is something entirely different from “the probability that the card is a king”. If I assume that’s just bad wording, and that you’re actually talking about the frequency of heads when some condition occurs, well now you’re doing frequentist probabilities, and we were talking about *epistemic* probabilities.
I’m not using the word irrelevant in the sense of “Doesn’t affect the probability calculation”, I’m using it in the sense of, “Doesn’t correspond to something that we care about”.
Yeah, I could have made my language clearer in my second paragraph. I was talking about the “probability of guessing the correct card” for a particular guessing strategy. And the probability of the next card being a king over some set of situations corresponds to the probability that the strategy of always guessing “King” for the next card gives the correct solution.
Anyway, my point was that you can manipulate your probability of being correct by changing which situations are included inside this calculation.