Write a sentence describing each of the following probabilities. (Note: A' = the complement of event A) P(A or A') = 1 P(A and A') = 0

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

Step-by-step explanation:

correct answer:

i^31 = -i