Union and intersection bounds

This is basic stuff but it can be very useful

• “at least one”: use a union, which is upper bounded by the summation of probabilities and lower-bounded by the single largest probability

• “all of them”: use an intersection, which is upper-bounded by the smallest probability. For the lower bound is a bit tricky. $P(A \cap B) = P(A) + P(B) - P(A\cup B)$. If $P(A) + P(B) < 1$, then we can make them maximally separate. If $P(A) + P(B) > 1$, then there must be some intersection. So $P(A \cap B) \geq P(A) + P(B) - \min(1, P(A) + P(B))$

Remember that “not one” is equivalent to “all of them” (the laws of set negation).