Answer:
The following properties hold for all events A, B.
• P(∅) = 0.
• 0 ≤ P(A) ≤ 1.
• Complement: P(A) = 1 − P(A).
• Probability of a union: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
For three events A, B, C:
P(A∪B∪C) = P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C).
If A and B are mutually exclusive, then P(A ∪ B) = P(A) + P(B).
• Conditional probability: P(A | B) = P(A ∩ B)
P(B)
.
• Multiplication rule: P(A ∩ B) = P(A | B)P(B) = P(B | A)P(A).
• The Partition Theorem: if B1, B2, . . . , Bm form a partition of Ω, then
P(A) = X
m
i=1
P(A ∩ Bi) = X
m
i=1
P(A | Bi)P(Bi) for any event A.
As a special case, B and B partition Ω, so:
P(A) = P(A ∩ B) + P(A ∩ B)
= P(A | B)P(B) + P(A | B)P(B) for any A, B.
• Bayes’ Theorem: P(B | A) = P(A | B)P(B)
P(A)
.
More generally, if B1, B2, . . . , Bm form a partition of Ω, then
P(Bj | A) = P(A | Bj )P(Bj)
Pm
i=1 P(A | Bi)P(Bi)
for any j.
• Chains of events: for any events A1, A2, . . . , An,
P(A1∩A2∩. . .∩An) = P(A1)P(A2 | A1)P(A3 | A2∩A1). . . P(An | An−1∩. . .∩A
Explanation: