**QUESTION**

1.

Question 1

**Prompt 1:** For each of the following scenarios,
find the requested probability. Assume the sets AA, BB, and CC are
events from the same sample space SS. (Hint: Venn diagrams may help
with the visualization, although they are not required to answer
the questions.) If P(A) = .4P(A)=.4, P(B^c) = .7P(Bc)=.7, and P(A ∩
B^c) = .2P(A∩Bc)=.2, find P(A ∩ B)P(A∩B)

1 point

2.

Question 2

**Prompt 1:** For each of the following scenarios,
find the requested probability. Assume the sets AA, BB, and CC are
events from the same sample space SS. (Hint: Venn diagrams may help
with the visualization, although they are not required to answer
the questions.)

If P(A) = 0.9P(A)=0.9 and P(B) = 0.9P(B)=0.9, what is the lower bound for P(A \cup B)P(A∪B).

1 point

3.

Question 3

**Prompt 2:** Three popular options on a certain
type of car are AA leather seats, BB a sunroof, and CC heated
seats. In the past, P(A) = 0.55P(A)=0.55 (i.e. 55% of the customers
have requested option A), P(B) = 0.45P(B)=0.45, P(C) = 0.4P(C)=0.4.
Furthermore, P(A ∩ B) = 0.25P(A∩B)=0.25, P(A ∩ C) = 0.2P(A∩C)=0.2,
P(B ∩ C) = 0.15P(B∩C)=0.15 and P(A ∩ B ∩ C) = 0.1P(A∩B∩C)=0.1. Find
the probability that a customer will ask for at least one of the
three options.

1 point

4.

Question 4

**Prompt 2:** Three popular options on a certain
type of car are AA leather seats, BB a sunroof, and CC heated
seats. In the past, P(A) = 0.55P(A)=0.55 (i.e. 55% of the customers
have requested option A), P(B) = 0.45P(B)=0.45, P(C) = 0.4P(C)=0.4.
Furthermore, P(A ∩ B) = 0.25P(A∩B)=0.25, P(A ∩ C) = 0.2P(A∩C)=0.2,
P(B ∩ C) = 0.15P(B∩C)=0.15 and P(A ∩ B ∩ C) = 0.1P(A∩B∩C)=0.1.

Find the probability that a customer will not ask for any of these three options.

1 point

5.

Question 5

**Prompt 2:** Three popular options on a certain
type of car are AA leather seats, BB a sunroof, and CC heated
seats. In the past, P(A) = 0.55P(A)=0.55 (i.e. 55% of the customers
have requested option A), P(B) = 0.45P(B)=0.45, P(C) = 0.4P(C)=0.4.
Furthermore, P(A ∩ B) = 0.25P(A∩B)=0.25, P(A ∩ C) = 0.2P(A∩C)=0.2,
P(B ∩ C) = 0.15P(B∩C)=0.15 and P(A ∩ B ∩ C) = 0.1P(A∩B∩C)=0.1.

Find the probability that a customer will ask for heated leather seats but not a sunroof.

1 point

6.

Question 6

**Prompt 2:** Three popular options on a certain
type of car are AA leather seats, BB a sunroof, and CC heated
seats. In the past, P(A) = 0.55P(A)=0.55 (i.e. 55% of the customers
have requested option A), P(B) = 0.45P(B)=0.45, P(C) = 0.4P(C)=0.4.
Furthermore, P(A ∩ B) = 0.25P(A∩B)=0.25, P(A ∩ C) = 0.2P(A∩C)=0.2,
P(B ∩ C) = 0.15P(B∩C)=0.15 and P(A ∩ B ∩ C) = 0.1P(A∩B∩C)=0.1.

Find the probability that a customer will ask for at most two of the options.

1 point

7.

Question 7

**Prompt 2:** Three popular options on a certain
type of car are AA leather seats, BB a sunroof, and CC heated
seats. In the past, P(A) = 0.55P(A)=0.55 (i.e. 55% of the customers
have requested option A), P(B) = 0.45P(B)=0.45, P(C) = 0.4P(C)=0.4.
Furthermore, P(A ∩ B) = 0.25P(A∩B)=0.25, P(A ∩ C) = 0.2P(A∩C)=0.2,
P(B ∩ C) = 0.15P(B∩C)=0.15 and P(A ∩ B ∩ C) = 0.1P(A∩B∩C)=0.1.

Find the probability that a customer will ask for exactly two of the options.

1 point

8.

Question 8

**Prompt 3:** A message of length 5 digits is to be
sent. Each digit can be a 0, 1, or 2.

What is the cardinality of the sample space?

1 point

9.

Question 9

**Prompt 3:** A message of length 5 digits is to be
sent. Each digit can be a 0, 1, or 2.

If every message is equally likely, what is the probability that the message consists of 2 zeros, 2 ones, and 1 two? Round your answer to have four decimal places.

1 point

10.

Question 10

**Prompt 3:** A message of length 5 digits is to be
sent. Each digit can be a 0, 1, or 2.

What is the probability that the message contains at least one zero? Round your answer to have three decimal places.