Question

2. Probability (30%). Figure out the probability in the following scenarios.

(a) A number generator is able to generate an integer in the range of [1, 100], where each number has equal chances to be generated. What is the probability that a randomly generated number x is divisible by either 2 or 3, i.e., P(2 | x or 3 | x)? (5%)

(b) In a course exam, there are 10 single-choice questions, each worthing 10 points and having 4 choices (A, B, C, and D, with only one correct). There is one student, denoted as s, who has learned nothing from the course and hence has to randomly guess the answers. That means for any question, each one of the four choices has equal chances to be picked up by s. What is the probability that s passes the exam (earning a total of ≥ 60 points)? (5%)

(c) Consider three positive integers, x1, x2, x3, which satisfy
the inequality below:

x1 +x2 +x3 =17. (1)

Let’s assume each element in the sample space (consisting of solution vectors (x1, x2, x3) satisfying the above conditions) is equally likely to occur. For example, we have equal chances to have (x1, x2, x3) = (1, 1, 15) or (x1, x2, x3) = (1, 2, 14). What is the probability the events x1 +x2 ≤8occurs,i.e.,P(x1 +x2 ≤8|x1 +x2 +x3 =17andx1,x2,x3 ∈Z+)(Z+ isthe set containing all the possible positive integers)? (5%)

(d) There are unlimited fake coins and only one real coin. The fake coins and the real coin are almost the same and can only be detected by a special machine. At the very beginning, there are two coins in a bag, one fake and the other real (but we don’t know which one is real). We continue the following process till the real coin is found: At the each step, we randomly sample one coin from the bag and examine whether it is fake. If yes, we put the coin back to the bag, additionally put in another fake coin, and randomly draw a coin for examination. The sampling process won’t stop until we find the real coin. Assuming that each coin (either fake or real) has equal chances to be selected, what is the probability that we sample 9 times but still cannot find the real coin (and hence has to continue the sampling process)? (5%)

(e) From a random sports news, the probability of observing the
word “*ball*” and “*player*” is 0.8 and 0.7,
respectively. For a non-sports news, the probability to observe
“*ball*” is 0.1, so does that to observe “*player*”.
Let’s assume that in any article, the appearance of any two words
(including “*ball*” and “*player*”) are independent
with each other. Also, the probability of sports news’ occurrences
is 0.2. Given a news report x containing both “*ball*” and
“*player*”, what is the probability that x is a sports news.
(10%)

Answer #1

Thus the required probability is

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**Note For other questions re post
them.**

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Consider three positive integers, x1, x2, x3, which satisfy the
inequality below:
x1 + x2 + x3 = 17.
Let’s assume each element in the sample space (consisting of
solution vectors (x1, x2, x3) satisfying the above conditions) is
equally likely to occur. For example, we have equal chances to have
(x1, x2, x3) = (1, 1, 15) or (x1, x2, x3) = (1, 2, 14). What is the
probability the events x1 + x2 ≤ 8 occurs, i.e., P(x1...

1. From a random sports news, the probability of observing the
word “ball” and “player” is 0.8 and 0.7, respectively. For a
non-sports news, the probability to observe “ball” is 0.1, so does
that to observe “player”. Let’s assume that in any article, the
appearance of any two words (including “ball” and “player”) are
independent with each other. Also, the probability of sports news’
occurrences is 0.2. Given a news report x containing both “ball”
and “player”, what is the...

Question 2
Given the following probability distribution, what is the
expected value?
Outcome
P(Outcome)
20
0.15
1
0.15
19
0.10
2
0.09
17
0.51
Round to 3 decimal places as needed.
Question 3
Brandybuck Insurance Company (BIC) is deciding whether to insure
the lives of those leading a quest to Moria. Based on past
experience, the probability of surviving such a quest is 83.4%. If
BIC charges a premium of 8,565 silver coins and would pay a death
benefit of...

In the following probability distribution, the random variable
x represents the number of activities a parent of a
6th to 8th-grade student is involved in. Complete parts (a)
through (f) below.
x
0
1
2
3
4
P(x)
0.395
0.075
0.199
0.195
0.136
(a) Verify that this is a discrete probability
distribution.
This is a discrete probability distribution because the sum of
the probabilities is ___and each probability is ___ (Less than or
equal to 1; Greater than or equal...

Multiple Choice
Select the best answer from the available choices for each
question.
Which of the following is NOT part of the definition of
a sample space S?
S can be discrete or continuous
Each outcome must be in S at most once
Each element in S is equally likely
Each outcome must be in S at least once
S is a set of possible outcomes in an experiment
Three A’s, three B’s, and two C’s are arranged at
random...

1. Usain Bolt sprints along the track, with his feet pushing
down and behind him. Therefore the track actually pushes Usain Bolt
along the track. Group of answer choices True or False
2.You study two objects in motion, Object J and Object S. Each
object has its own velocity graph, with same time and velocity
scales. The slope of each graph is positive, upward and to the
right ⁄ but Object J's tilt angle is only 10º and Object
S's tilt...

Please answer the following Case
analysis questions
1-How is New Balance performing compared to its primary rivals?
How will the acquisition of Reebok by Adidas impact the structure
of the athletic shoe industry? Is this likely to be favorable or
unfavorable for New Balance?
2- What issues does New Balance management need to address?
3-What recommendations would you make to New Balance Management?
What does New Balance need to do to continue to be successful?
Should management continue to invest...

In February 2012, the Pepsi Next product was launched into
the US market. This case study provides students with an
interesting insight into PepsiCo’s new product process and some of
the challenging decisions that they faced along the way.
Pepsi Next Case Study
Introduction
Pepsi Next was launched by PepsiCo into the US market in
February 2012, and has since been rolled out to various
international markets (for instance, it was launched in Australia
in September 2012).
The new product...

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