Question

Q1:

a) Re-derive the inclusion-exclusion principle for two events using only the probability axioms.

Probability axioms: Given an event A in Ω:

A1) P(A) >= 0

A2) P(Ω) = 1

A3) P(U (from i=1 to n) A_i) = Σ (from i=1 to n) P(A_i) - if A_i's are disjoint/ mutually exclusive

Inclusion Exclusion Principle for two events: (A U B) = (A) + (B) + (A ∩ B)

b) Then, using only the axioms and the inclusion-exclusion principle for two events, derive the formula for the inclusion-exclusion principle for three events.

Inclusion Exclusion Principle for three events: (A U B U C) = (A) + (B) + (C) - (A ∩ B) - (A ∩ C) - (B ∩ C)

Answer #1

Prove using only the axioms of probability that if A and B are
events and A ⊂ B, then P(Ac ∩ B) = P(B) − P(A).

Distributions of Distinct Objects to Distinct Recipients
and using the principle of inclusion-exclusion:
Let X={1,2,3,...,8 } and Y={a,b,c,d,e}.
a) Count the number of surjections from X to Y.
b) Count the number of functions from X to Y whose image consists
of exactly three elements of V.

1)Three independent reviewers are reviewing a book. let
A1 denote the event that a favorable review is submitted
by reviewer, I = 1, 2, 3. Assume that A1, A2,
and A3 are mutually independent and that
P(A1) = 0.6, P(A2) =0.57, and
P(A3) = 0.4.
a) Compute the probability that at least one of the reviewers
submit a favorable review.
b) Compute the probability that exactly two reviewers submit
favorable reviews.

A computer consulting firm presently has bids out on three
projects. Let Ai = {awarded project i}, for i = 1, 2, 3, and
suppose that P(A1) = 0.22, P(A2) = 0.25, P(A3) = 0.28, P(A1 ∩ A2) =
0.14, P(A1 ∩ A3) = 0.04, P(A2 ∩ A3) = 0.06, P(A1 ∩ A2 ∩ A3) = 0.01.
Express in words each of the following events, and compute the
probability of each event.
(a) A1 ∪ A2 Express in words the...

A computer consulting firm presently has bids out on three
projects. Let Ai = {awarded project i}, for i = 1, 2, 3,
and suppose that P(A1) = 0.22, P(A2) = 0.25,
P(A3) = 0.28, P(A1 ∩ A2) = 0.13,
P(A1 ∩ A3) = 0.03, P(A2 ∩
A3) = 0.07, P(A1 ∩ A2 ∩
A3) = 0.01.
Express in words each of the following events, and compute the
probability of each event.
a) A1 ∪ A2
Express in words the...

Q.3. (a) Let an experiment consist of tossing two standard
dice. Define the events, A = {doubles appear} (That is (1, 1), (2,
2) etc..)
B = {the sum is bigger than or equal to 7 but less than or
equal to 10}
C = {the sum is 2, 7 or 8}
(i) Find P (A), P (B), P (C) and P (A ∩ B ∩ C). (ii) Are
events A, B and C independent?
(b) Let the sample space...

A certain system can experience three different types of
defects. Let Ai (i = 1,2,3) denote the
event that the system has a defect of type i. Suppose that
the following probabilities are true.
P(A1) =
0.16 P(A2)
=
0.10 P(A3)
= 0.08
P(A1 ∪ A2) =
0.18 P(A1
∪ A3) = 0.19
P(A2 ∪ A3) =
0.14 P(A1
∩ A2 ∩ A3) = 0.02
(a) What is the probability that the system does not have a type
1 defect?
(b) What is the...

Part 1
The three most popular options on a certain type of new car are
a built-in GPS (A), a sunroof (B), and an automatic transmission
(C). If 48% of all purchasers request A, 59% request B, 74% request
C, 68% request A or B, 85% request A or C, 83% request B or C, and
90% request A or B or C, determine the probabilities of the
following events. [Hint: "A or B" is the event that at least...

A certain system can experience three different types of
defects. Let Ai (i = 1,2,3) denote the
event that the system has a defect of type i. Suppose that
the following probabilities are true.
P(A1) =
0.11
P(A2) =
0.07
P(A3) = 0.05
P(A1 ∪
A2) = 0.15
P(A1 ∪
A3) = 0.14
P(A2 ∪
A3) = 0.1
P(A1 ∩
A2 ∩ A3) = 0.01
(Round your answers to two decimal places.)
(a) Given that the system has a type...

4. (Sec 2.5) Consider purchasing a system of audio components
consisting of a receiver, a pair of speakers, and a CD player. Let
A1 be the event that the receiver functions properly throughout the
warranty period, A2 the event that the speakers function properly
throughout the warranty period, and A3 the event that the CD player
functions properly throughout the warranty period. Suppose that
these events are (mutually) independent with P(A1) = .95, P(A2) =
.98 and P(A3) = .80....

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