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:

the most boring thing ever written

step-by-step explanation:

i am writing something. yes, i plan to make it the most boring thing ever written. i go to the store. a car is parked. many cars are parked or moving. some are blue. some are tan. they have windows. in the store, there are items for sale. these include such things as soap, detergent, magazines, and lettuce. you can enhance your life with these products. soap can be used for bathing, be it in a bathtub or in a shower. apply the soap to your body and rinse. detergent is used to wash clothes. place your dirty clothes into a washing machine and add some detergent as directed on the box. select the appropriate settings on your washing machine and you should be ready to begin. magazines are stapled reading material made with glossy paper, and they cover a wide variety of topics, ranging from news and politics to business and stock market information. some magazines are concerned with more recreational topics, like sports card collecting or different kinds of hairstyles. lettuce is a vegetable.

step-by-step explanation:

options