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[Combinatorics and Probability] Week 4 Probability: Inclusion-Exclusion Formula 본문
[Combinatorics and Probability] Week 4 Probability: Inclusion-Exclusion Formula
yjyuwisely 2024. 3. 21. 07:00Inclusion-Exclusion Formula
It is known that two events A and B in some probability space have probabilities 0.7 and 0.8. What is the minimal possible probability of an event "A and B" (the intersection of both events)?
0.7 + 0.8 - 1 = 0.5
Recall the inclusion and exclusion formula: Pr[A or B] = Pr[A] + Pr[B] - Pr[A and B].
This probability could not exceed 1, and Pr[A] + Pr[B] = 1.5, so Pr[A and B] should be at least 0.5.
This is indeed possible (if A and B together cover the entire space)
The formula that relates the probabilities of A, B, and their intersection is:
P(A ∩ B) = P(A) + P(B) - P(A ∪ B)
Since P(A ∪ B) cannot be greater than 1 (the total probability space),
the minimum value for P(A ∩ B) would be achieved when P(A ∪ B) is at its maximum, which is 1. So we get:
Minimum P(A ∩ B) = P(A) + P(B) - 1
0.7 + 0.8 - 1 = 0.5
It is known that two events A and B in some probability space have probabilities 0.7 and 0.8. What is the maximal possible probability of an event "A and B" (the intersection of both events)?
0.7
This happens when A is a part of B (and this is the maximal possible probability, since "A and B" is a part of A).
