“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.
“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.