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1) The probability of any event is always a number between 0 and 1, inclusive.
2) For two events A and B, P(A ∪ B) is always equal to P(A) + P(B).
3) Conditional probability measures the probability of one event given that another has occurred.
4) The Multiplication Rule says P(A ∩ B) = P(A)·P(B|A).
5) If two events are independent, then knowing one occurred changes the probability of the other.
6) Bayes’ Theorem is used to update probabilities when new information becomes available.
7) A random variable always takes on values that are unpredictable, but each value has a probability distribution.
8) The expected value of a discrete random variable is essentially its long-run average outcome.
9) For mutually exclusive events A and B, both can occur together with positive probability.
10) Disjoint (mutually exclusive) events are always independent.
11) If A ⊆ B and P(B)>0, then P(A|B) = P(A)/P(B).
12) Complement rule: P(Ac) = 1 − P(A).
13) If P(A|B) = P(A) (with P(B)>0), then A and B are independent.
14) For any events A and B, P(A ∩ B) > min{P(A), P(B)}.
15) If A and B are mutually exclusive and both have positive probability, then they cannot be independent.
16) The conditional probability P(A|B) is undefined when P(B) = 0.
17) Union bound: For any events A and B, P(A ∪ B) = P(A) + P(B).
18) For a discrete random variable X with PMF p(x), we must have ∑x p(x) = 1.
19) A highly accurate medical test always implies that a positive result means the person almost surely has the disease, regardless of disease prevalence.