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:

answer

a,b, and d

explanation

a reflection angle is more than

but less than;

an angle that measures

is a reflex angle.(true)

an angle that measures less than

is acute.(true)

provided we are considering angles greater or equal to zero.

an acute angle lies in the range.

an obtuse angle is less than 180° but more than 90°

an angle that measures 300° is obtuse (false)

an angle that measures 65° is an acute angle.(true)