I'm trying to understand what the sample (Kolmogorov, elementary event) space and event (Kolmogorov, random event) space is for the following problem. It has a solution which someone has presented, however I'm trying to put it in terms of above. I think some of my confusion if these are mutually exclusive events, $P(A) + P(B)$ cannot add up to more than $1$. Can anyone help here?
Suppose we have the following information:
- There is a 60 percent chance that it will rain today.
- There is a 50 percent chance that it will rain tomorrow.
- There is a 30 percent chance that it does not rain either day.
Find the following probabilities:
- The probability that it will rain today or tomorrow.
- The probability that it will rain today and tomorrow.
- The probability that it will rain today but not tomorrow.
- The probability that it either will rain today or tomorrow, but not both.
A is the event it will rain today and B is the event it will rain tomorrow.
- $P(A) = 0.6$
- $P(B) = 0.5$
- $P(A^c \cap B^c) = 0.3$