Conrad wrote:

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.

I think it would be worthwhile to examine this conclusion - as it might seem to be an obvious one to a lot of people. Let us assume that there is a very good mechanical arm that makes a completely fair toss of the coin in the opinion of all humans so that we can talk entirely about the bias of the coin.

Let's say that the mechanism makes one toss; all you know is that the coin is biased - not how. Assume that it comes up heads; what does this tell you about the bias? Conrad asserts that it will certainly be biased in favor of heads. How much? Will it always show up as heads? 3 times out of 4? As it turns out, you have no way of knowing.

It could be that it is in fact only 1/3 biased towards heads; then it would be much wiser to bet on tails in the future, no? It could be that it is actually 100 times more likely that tails will come up; you simply can't tell the difference from the first toss.

So let's consider more coin tosses. What if it comes up heads once and then tails 5 times in a row? Could you tell me exactly what the bias is? Is it 5/6 towards tails perhaps? What about 50 tails and 15 heads? In fact, it is still not possible to say anything at all about what the bias is.

Since you probably have a heuristic method of analysis (intuition) you will in time see which side is the best bet; i.e. you'll conclude which side is most likely to be biased and you'll probably be correct - with higher accuracy as the amount of tosses increase. However; there is no logic, rationalism or deduction in the world that could tell you exactly what the bias is. This is true after any integer amount of coin tosses.