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If I am given that on Saturday there is 0.6 probability of raining and 0.7 probability of raining on Sunday, then there is a 0.88 chance that it will rain at least one of the days (this is using independence assumption). However, what if nothing about their probability structures were known? Ie. they could be negatively correlated. They could be correlated with $r=1$. What would be the bounds on the probability of it raining at least one day? Im thinking $0.7,0.88$?

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Let $A$ be the event it rains on Saturday, and $B$ the event it rains Sunday. You want bounds on $\Pr(A\cup B)$.

It is clear that $\Pr(A\cup B)\le 1$. We cannot do better. For example, the rain god could choose an integer at random between $1$ and $10$. If the number is between $1$ and $6$, it rains on Saturday. If it is between $4$ and $10$, it rains on Sunday. It is clear that it rains for sure on at least one of Saturday or Sunday.

The best lower bound is $0.7$. We can't find a smaller one, since $\Pr(A\cup B)\ge \Pr(B)$. Suppose that every time it rains on Saturday, it rains on Sunday. (In set-theoretic language, we have $A\subset B$). Then the probability of rain on at least one day is $0.7$.

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