I have read through many of the threads in StackExchange, but I cannot seem to grasp the idea of a 3 event union fully. I have these probabilities to use: $$P(A) = \frac{73}{250}, P(B)={\frac{9}{25}}, P(C)={\frac{13}{250}}$$$$ P(A\cap B)={\frac{171}{1000}},P(A\cap C)={\frac{29}{1000}},P(B\cap C)={\frac{1}{1000}}$$ I would have to calculate the union for all groups, and I searched the formula from previous examples: $$P(A \cup B \cup C ) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C).$$ It is basic adding and substracting for the first 6 parts, but I'm very confused about the last part about the intersection of A, B and C. I have tried to reverse the formula, but it doesn't seem possibly without knowing the union of these groups first.
The boundaries for the union for all the three groups is also something I do not fully grasp. If there is a certain probability for the group to happen, doesnt it mean there are no lower or upper boundaries to it? What makes the probability different for these cases? I tried to figure out an answer via different sources from the internet, and I ended up with the following: $$P(A \cup B \cup C ) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C)=\frac{503}{1000} $$ However, this formula does not have the addition of the three groups and their intersections, and yet might work as a lower boundary. Perhaps the upper boundary would be something added to this, such as the intersection of groups A, B and C, but then I end up with the same problem I started with.
How do the boundaries of probability work with set theory like this?
The answer is $$\frac{503}{1000}+\Pr(A\cap B\cap C)$$ and we don't know what $\Pr(A\cap B\cap C)$ is. However, we do know that $$0\leq\Pr(A\cap B\cap C)\leq\Pr(B\cap C)=\frac1{1000},$$ so $$ \frac{503}{1000}\leq\Pr(A\cup B\cup C)\leq \frac{504}{1000}$$ is the most we can say.