Are there 2 (or more) types of probabilities and I am just mixing them up
Yes, there are conditional probabilities and unconditional probabilities.
The unconditional probability of 21 heads in a row is 0.5^21[1]. The conditional probability of 21 heads in a row given that the first 20 were all heads is 0.5.
Conditional probability is just a division: the conditional probability of some event A given that B happened is just the unconditional probability of both A and B divided by the probability of B. In symbols: P(A | B) = P(A & B) / P(B).
As is common, this assumes that the coin flips are independent of one another. An alternative might be that the coin was flipped “lazily” such that it more often shows the same face as the previous flip, but over the long run still flips 50% heads. A “properly” flipped coin should not depend upon the results of any or all previous flips.
Yes, there are conditional probabilities and unconditional probabilities.
The unconditional probability of 21 heads in a row is 0.5^21[1].
The conditional probability of 21 heads in a row given that the first 20 were all heads is 0.5.
Conditional probability is just a division: the conditional probability of some event A given that B happened is just the unconditional probability of both A and B divided by the probability of B. In symbols: P(A | B) = P(A & B) / P(B).
Bayes’ Law comes from simple algebra on this.
As is common, this assumes that the coin flips are independent of one another. An alternative might be that the coin was flipped “lazily” such that it more often shows the same face as the previous flip, but over the long run still flips 50% heads. A “properly” flipped coin should not depend upon the results of any or all previous flips.