We can understand it better by taking the extreme example. I will be much more surprised to hear that the coin was tails and that we are now at day #500,000, then that the coin was heads and that it is the first day.
You should not be more surprised in that situation. The more days there are, the more that the extra tails awakenings push down the probability of heads. With 500000 awakenings, the probability gets pushed down by a lot. Now heads is 1⁄500001 per-awakening probability, same as tails-day-1 and tails-day-500000
You are claiming that if I will be wake up 500,000 times if the coin lands tails, I should be virtually certain a priori that the coin will land tails. I am not; I would not be surprised at all if it landed heads. In fact, as I have been saying, the setup does not make me expect tails in any way. So at the start the probability remains 50% heads, 50% tails.
I do not. I mean reporting my opinion when someone asks, “Do you think the coin landed, heads, or tails?” I will truthfully respond that I have no idea. The fact that I would be woken up multiple times if it landed tails, did not make it any harder for the coin to land heads.
I’d recommend distinguishing between the probability that the coin landed heads (which happens exactly once), and the probability that, if you were to plan to peak you would see heads (which would happen on average 250,000 times).
The problem is that you are counting frequencies, and I am not. It is true that if you run the experiment many times, my estimate will change, from the very moment that I know that the experiment will be run many times.
But if we are going to run the experiment only once, then even if I plan to peek, I would expect with 50% probability to see heads. That does not mean “per awakening” or any other method of counting. It means that if I saw heads, I would say, “Not surprising; that had a 50% chance of happening.” I would not say, “What an incredible coincidence!!!!”
You should not be more surprised in that situation. The more days there are, the more that the extra tails awakenings push down the probability of heads. With 500000 awakenings, the probability gets pushed down by a lot. Now heads is 1⁄500001 per-awakening probability, same as tails-day-1 and tails-day-500000
You are claiming that if I will be wake up 500,000 times if the coin lands tails, I should be virtually certain a priori that the coin will land tails. I am not; I would not be surprised at all if it landed heads. In fact, as I have been saying, the setup does not make me expect tails in any way. So at the start the probability remains 50% heads, 50% tails.
Yes, I am (assuming you mean per-awakening certainty).
I do not. I mean reporting my opinion when someone asks, “Do you think the coin landed, heads, or tails?” I will truthfully respond that I have no idea. The fact that I would be woken up multiple times if it landed tails, did not make it any harder for the coin to land heads.
I’d recommend distinguishing between the probability that the coin landed heads (which happens exactly once), and the probability that, if you were to plan to peak you would see heads (which would happen on average 250,000 times).
The problem is that you are counting frequencies, and I am not. It is true that if you run the experiment many times, my estimate will change, from the very moment that I know that the experiment will be run many times.
But if we are going to run the experiment only once, then even if I plan to peek, I would expect with 50% probability to see heads. That does not mean “per awakening” or any other method of counting. It means that if I saw heads, I would say, “Not surprising; that had a 50% chance of happening.” I would not say, “What an incredible coincidence!!!!”