There is one thing I don't understand about probabilities:
If we toss a coin, there is a 50% chance that it shows heads or tails. If we do it 20 times and all of them showed heads, there is still a 50% chance that the next one shows heads, since the tosses are independent. However, we also know that series of X tosses showing heads are increasingly improbable when X grows. So, although there is a 50% chance that the toss shows heads again, at the same time the probability that it shows heads again are lower.
Why do we have to take into account one piece of information and not the other one when finding the probability that the next toss will show heads or tails? Are there 2 (or more) types of probabilities and I am just mixing them up (I'm thinking on things like the reported "probabilities" that polls show about one party or another getting elected in an election, for example)? Is the difference related to ergodicity (time vs ensemble averages)?