Probability after a series of outcomes - what is the formula?

Question

Probability theory

If it rains for 3 days, does the chance of rain on the 4th day increase or decrease?
Same task in a different way. If Scheherizade successfully saved her life by telling a fairy tale for a thousand nights in a row, on the 1001st night, do the chances of survival increase or decrease?
Is there some formula or concept in probability theory that allows you to calculate such chances?

Answer 1

In these examples, the probabilities will change due to the changed conditions, not because of what the outcome was before.
A simple example with constant conditions:
We toss a coin and try to guess heads or tails.
No matter how many times we toss (1 or 1000) the probability of heads or tails next time is the same as the first.
An example with changing conditions:
There are 100 coins in a box (50 pieces of 1-kopeck and 2-kopeck coins each)
We pull out 1 coin, after which we add a new, but 3-kopeck coin to the box in its place.
So the probability that we will pull out 1-kopeck coin 1 time will be 0.5, but the more often we get such coins, the less this probability will be.