“Suppose our information about bias in favour of heads is equivalent to our information about bias in favour of tail. Our pdf for the long-run frequency will be symmetrical about 0.5 and its expectation (which is the probability in any single toss) must also be 0.5. It is quite possible for an expectation to take a value which has zero probability density.”
What I said: if all you know is that it’s a trick coin, you can lay even odds on heads.
“We can refuse to believe that the long-run frequency will converge to exactly 0.5 while simultaneously holding a probability of 0.5 for any specific single toss in isolation.”
Again what I said: if the question is, “This is a trick coin: I’ve rigged it. I have written down here the probability that it’ll come up heads. Do you accept that the number I’ve written down is .5?”—You’ve got to say no. Since they’ve just told you it was rigged.
And if what they’ve written down is .50000000000001 and come back at you for it, then they stretched a point to say it was rigged.
So your problem is you haven’t grounded the example in terms of what we’re being asked to do.
Again, what difference does it make?
Conrad.
ps—Ofc, knowing, or even just suspecting, the coin is rigged, on the second throw you’d best bet on a repeat of the outcome of the first.
C.
Maybe I’m stupid here… what difference does it make?
Sure, if we had a coin-flip-predicting robot with quick eyes it might be able to guess right/predict the outcome 90% of the time. And if we were precognitive we could clean up at Vegas.
In terms of non-hypothetical real decisions that confront people, what is the outcome of this line of reasoning? What do you suggest people do differently and in what context? Mark cards?
B/c currently, as far as I can see, you’re saying, “The coin won’t end up ‘heads or tails’—it’ll end up heads, or it’ll end up tails.” True but uninformative.
Conrad.
ps—The thought experiment with the trick coin is ungrounded. If I’m being asked to lay even odds on a dollar bet that the coin is heads, then that’s rational—since the coin could be biased for heads, or tails (and the guy proposing the bet doesn’t know the bias). If I’m being asked to accept or reject a number meant to correspond to the calculated or measured likelihood of heads coming up, and I trust the information about it being biased, then the only correct move is to reject the 0.5 probability. It has nothing to do with frequentist, Bayesian, or any other suchlike.
C.