Question

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 one of the two options is requested; try drawing a Venn diagram and labeling all regions.]

(a) The next purchaser will request at least one of the three options.

(b) The next purchaser will select none of the three options.

(c) The next purchaser will request only an automatic transmission and not either of the other two options.

(d) The next purchaser will select exactly one of these three options.

Part 2

A certain system can experience three different types of
defects. Let *A _{i}* (

*P*(*A*_{1}) =
0.14 *P*(*A*_{2})
=
0.10 *P*(*A*_{3})
= 0.07

*P*(*A*_{1} ∪ *A*_{2}) =
0.16 *P*(*A*_{1}
∪ *A*_{3}) = 0.17

*P*(*A*_{2} ∪ *A*_{3}) =
0.14 *P*(*A*_{1}
∩ *A*_{2} ∩ *A*_{3}) = 0.02

(a) What is the probability that the system does not have a type
1 defect?

(b) What is the probability that the system has both type 1 and
type 2 defects?

(c) What is the probability that the system has both type 1 and
type 2 defects but not a type 3 defect?

(d) What is the probability that the system has at most two of
these defects?

Answer #1

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2. The three most popular options on a certain type of new car
are an automatic transmission (A), a sunroof (B), and a built-in
GPS (C). If 70% of all purchasers request A, 80% request B, 75%
request C, 85% request A or B, 90% request A or C, 95% request B or
C, and 98% request A or B or C, determine the probabilities of the
following events. [Hint: “A or B” is the event that at least one...

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 37% of all purchasers
request A, 46% request B, 63% request C,
53% request A or B, 71% request A or
C, 74% request B or C, and 77% request
A or B or C, determine the probabilities
of the following events. [Hint:"A or B"
is the event that at least one of the...

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...

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...

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) = 0.06
P(A3) = 0.04
P(A1 ∪
A2) = 0.14
P(A1 ∪
A3) = 0.13
P(A2 ∪
A3) = 0.08
P(A1 ∩
A2 ∩ A3) = 0.01
(Round your answers to two decimal places.)
(a) Given that the system has a type...

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 P(A1) = 0.25, P(A2) = 0.29, P(A3) = 0.33,
P(A1 ∪ A2) = 0.5, P(A1 ∪ A3) = 0.53, P(A2 ∪ A3) = 0.54,
P(A1 ∩ A2 ∩ A3) = 0.02
(a) Find the probability that the system has exactly 2 of the 3
types of defects.
(b) Find the probability...

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) = 0.23,
P(A2) = 0.25,
P(A3) = 0.29,
P(A1 ∩ A2) =
0.09,P(A1 ∩ A3) =
0.11, P(A2 ∩ A3) =
0.07, P(A1 ∩ A2 ∩
A3) = 0.02. Use the probabilities given above
to compute the following probabilities. (Round your answers to four
decimal places.)
(a) P(A2 |
A1) =
(b) P(A2 ∩...

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...

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.

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