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:

me

step-by-step explanation:

c.   x=2.4cm

step-by-step explanation:

similar triangles are triangles which have the same shape but not the same size. this means their angle measures are equal but their side lengths are not instead they are proportional. to solve, we will set up a proportion and solve.

a proportion is two equal ratios set equal to each other. we form the ratios by finding corresponding sides (sides which match to each other on both triangles by their position). out ratios will be one side of the small triangle over the corresponding side on the big triangle as shown below:

.

to solve the proportion, we'll cross multiply by multiplying numerator with denominator across the equal sign.