Range of probability for non-independent events

Suppose the probability of rain on day 1 is $$p_1$$ and probability of rain on day 2 is $$p_2$$. Then the probability of rain for the entire two-day period is $$1-(1-p_1)(1-p_2)$$, under the assumption that the two days are independent.

Now my question is: if the independence assumption does not hold true, the probabilities cannot be simply multiplied together and the probability of rain for the two-day period could potentially take on different values depending on the level of dependence between the two days. In this case, what would be a range for the probability of rain during the two-day period?

I really do not have much of a clue to this one. Any help is much appreciated. Thanks!

First of all, the probability of at least one of the events happening is the probability of the union and it is given by: $$p(A\cup B)=p_A+p_B-p(A\cap B)$$ This equality can be easily seen on a Venn diagram. Now, we also know that:
$$0\leq p(A\cap B)\leq \min(p_A, p_B)$$
Substituting on the previous expression, we get: $$\max(p_A,p_B)\leq p(A\cup B)\leq p_A+p_B$$ Notice that the upper bound is achieved is $$A$$ and $$B$$ are disjoint (no overlap) and the lower bound is achieved if $$A\subset B$$ or $$B\subset A$$ (full overlap). Any other case shall fall in between these bounds.
• If I understand, you want to know the probability of rain on the whole period based on the probabilities for each day. Is that right? If so, you need information about how one event impact the probability of the other - more precisely, you need to know $P(A\cap B)$ (or equivalently $P(A|B)$, or $P(B|A)$). Without this information and without assuming independence, $P(A\cup B)$ can take any value between $\max(p_A, p_B)$ and $p_A+p_B$. Dec 3, 2019 at 19:20