probability notation

MGT 233 – Homework #2
Spring 2018
This homework assignment is due at the beginning of class on Monday, October 8, 2018.
Instructions: Use proper probability notation for all parts of each problem. Show all work in
order to receive full credit. For all normal distribution, include a graph indicating the mean and
shade the area under investigation. For your own sake, be neat.
Problem #1
Alanarnette.com- Through June of 2017, nearly 5,000 people
have summited Mount Everest, located in Nepal. Roughly
25% of the individuals that have summited the highest peak
in the world (above sea level) are Sherpas, local people who
are hired and paid to help climbers carry tents and cook food
at high camps on the summit.
In a sample of 15 individuals who have summited Mount
Everest:
a. How many people do you expect, of those surveyed, to be Sherpas?
b. Compute the variance and standard deviation of Sherpas that have summited the peak.
c. What is the probability that ten of these individuals are Sherpas?
d. What is the probability that six or fewer of these individuals are Sherpas?
e. What is the probability that at least thirteen of these individuals are Sherpas?
f. What is the probability that ten, eleven, or twelve of these individuals are Sherpas?
g. Suppose the rate of individuals who hire Sherpas decreases. Now, only 17% of the
individuals who summit the mountain are Sherpas. What is the probability that of the 17
people sampled, exactly 6 of these individuals are Sherpas? (Calculate using the
equation)
Problem #2
National Singles Day is a day for everyone, singles and
couples alike, to recognize our single friends, family
members, co-workers and fellow citizens. The U.S. Census
Bureau reports that 45% of U.S. residents age 18+ are
single, that’s 110.6 million U.S. taxpayers. In 2018,
National Singles Day is Saturday, September 22.
Assume that the samples below include citizens of The U.S.
a. In a sample of 10 individuals, what is the probability that exactly 4 claim they are single?
b. In a sample of 12 individuals, what is the probability that at least 8 claim they are single?
c. In a sample of 15 individuals, what is the probability that no more than 10 claim they are
single?
d. In a sample of 9 individuals, what is the probability that 3, 4, 5, or 6, claim they are
single?
Problem #3
worldpopulationreview.com – According to the US Census
Bureau’s population clock, the estimated 2018 United States
population (February 2018) is 327.16 million. By population, the
United States of America is the third largest country in the
world, falling far behind China (1.4 billion) and India (1.25
billion). Over 350 languages are used by the U.S. population in
which the four languages spoken at home by the most people.
The most commonly used language is English with used by 79.29% of population. Spanish is the
second most common language in the country with 12.85%. There is 0.64% using Chinese
language, and 7.22% use other languages (Tagalog, Vietnamese, French, Korean, Arabic, and
German)
Use EXCEL binom.dist.range functions to solve the following probabilities. Show proper
probability notation, show the Excel formula used, and carry your work to 5 decimal places.
a) In a sample of 10 expenditures, what is the probability exactly 4 speak Spanish at home?
b) In a sample of 12 Americans, what is the probability that at least 3 speak English at home?
c) In a sample of 15 Americans, what is the probability that 8 or fewer speak Chinese at home?
d) In a sample of 19 Americans, what is the probability that 3 to 9 do not speak English, Spanish
or Chinese at home?
e) Using the binom.inv function, when 18 Americans are sampled, there is a 95% probability
that less than or equal to how many were spoken Spanish at home?
Problem #4
Himanchal.org, nepalconnection.org.np –
The Himanchal Education Foundation is
a non-profit organization that exists to
promote and advance Mahabir Pun’s
vision for extended educational
opportunities in rural Nepal. HEF
originally was founded to support growth
of the Nangi school in Nepal. HEF also
supports economically sustainable
business opportunities and a computer
network in Nangi and surrounding
villages. Our goal remains to improve the health and life situations for villagers in rural Nepal.
Dr. Mahabir Pun earned bachelor and master degrees from UNK. The University of Nebraska
Kearney also awarded him with a doctorate of letters for his great accomplishments to advance
life in rural Nepal.
To help fund schools and health clinics in the mountains of rural Nepal, Dr. Pun opened a
wildly successful restaurant in Kathmandu, the capital of Nepal. During trekking/tourist
season, the restaurant averages 66 customers during the evening peak hours between 6:00pm
and 9:00pm.
Use the tables in the textbook for the probabilities.
a) How many customer visits are expected in 15 minutes?
b) What is the probability of no customers in 6 minutes?
c) What is the probability of 5 or fewer customers in 12 minutes?
d) What is the probability of 7 or more customers in 18 minutes?
e) What is the probability of 6, 7, 8, or 9 customers in 30 minutes?
Problem #5
The Trekking Agencies’ Association of Nepal (TAAN), launched new treks which will benefit
the schools in and around Nangi village. Mahabir Pun has always had the vision of a trekking
route close to home. This dream became a reality when the Borderlands Resort’s Director Megh
Ale agreed to team with the Himanchal Educational Foundation to create this new route.
These treks focus on using eco-lodges, conservation, and
environmentally sustainable travel. Trekkers will travel for
seven days to six villages, whose elevations range from 2,700
feet at the starting point in Beni along the Kali Gandaki River
to over 9,000 on the Mohare Ridge. On a clear day you can
see over 30 snow clad mountains including three mountains
over eight thousand meters (26,000 feet). For a comparison,
Pikes Peak in Colorado is 14,114 feet (8,075 feet above
Colorado Springs).
In addition to the grandeur of Nepal, these treks are eco-friendly and the lodges are owned and
operated by their local schools: all proceeds after expenses go back to the schools for educational
purposes and lodges are run by local villagers.
October and November are peak trekking season in Nepal, average 890 trekkers visited protected
areas each week. But the low season (December and January), on average, only 21 trekkers
visited protected areas each week.
a) How many trekkers visit protected areas are expected in 28 days in the low season?
b) What is the probability of 6 trekkers or fewer visited in three days in the low season?
c) What is the probability of more than 7 trekkers visited in one day in the low season?
d) What is the probability of 15 to 19 visits to the website in five days in the low season?
Eco-Trekking Lodge in Nangi, Nepal
Problem #6 –
Tourism.gov.np- Roughly 940,218 people traveled
to Nepal in 2017. During this year, tourism numbers
increased by about 25%, the largest increase in the
country’s history. The government sees the tourism
and hospitality industry to be a significant
opportunity to improve the economy of the country.
Texting with a mobile phone is now a required
amenity even for remote rural areas. Assuming
Nepal Wireless processes 6,120 text messages per hour during peak times in the rural town of
Nangi, Nepal. In Nangi, during peak times:
a) how many text messages are expected to be processed in 5 seconds?
b) what is the probability that no text messages are sent in 3 seconds?
c) what is the probability that 7 or fewer text messages are sent in 4 seconds?
d) what is the probability that 5 or more text messages are sent in 5 seconds?
e) What is the probability that 17 to 24 text messages are sent in 10 seconds?
f) What is the probability that exactly 8 text messages are sent in 2 seconds?
Problem #7
NepalWireless.net – The following story is told by Dr. Mahabir Pun, the Team Leader of Nepal Wireless
Networking Project:
The Beginning
It started with a wish. In 1996. I wished to
get Internet in my village for the first time after
Himanchal High School got four used computers
as presents from the students of a school in
Australia. Internet and e-mail were quite new terms
then.
Turning Point
I kept on asking people for ideas. I also wrote a very short e-mail to the BBC in 2001,
asking if they knew anybody who could give me ideas (if there were any) to get a cheaper
Internet connection to my remote village in Nepal. They
took my interview and wrote the articles “Village in the
Clouds Embraces Computers” and “Praise for
‘Inspirational’ Web Pioneer” about my school and the
computers we had built in wooden boxes. That article
changed everything: I got many responses with ideas
from people all over the world. That was the first time I
heard about Wi-fi (802.11b) wireless technology.
The Network Now
Starting in 2002, it has been already 14 years
since Nepal Wireless is working in to bring broadband Internet in the rural areas. Our focus is to
bring broadband Internet in the rural areas where no commercial Internet Service Providers
would like to go. The ultimate goal is to maximize the benefit of the technology to the rural
population. Nepal Wireless has built a network spanning 15 districts of Nepal. Altogether, 175
villages are now connected.
The Investigation
The Nepal Wireless network continually updates the
technology while expanding connection to rural mountain
villages. Of concern, is the network capacity at the Harmi
Relay tower? On average, during peak times the Harmi Relay
receives 55 megabytes of digital information per second with
a standard deviation of 15 megabytes per second. The Harmi
Relay can handle up to 85 megabytes of digital information
per second before the system starts buffering. When more
than 97 megabytes are received in 1 second, the tower starts
dropping data packets. Assume the data arrivals to the Harmi
Relay are random and normally distributed.
During peak times:
a) What is the probability that the Harmi Relay will receive no more than 49 megabytes in one
second?
b) What is the probability that the Harmi Relay will receive between 22 and 67 megabytes in
one second?
c) In a one second period, what is the probability that the Harmi Relay will need to buffer
information?
d) In any one second, what is the probability that the Harmi Relay will start dropping packets?
Problem #8
Bhutan is the one of five countries which The
Himalayas is spread across. It is the last great
Himalayan kingdom, shrouded in mystery and
magic, where a traditional Buddhist culture
carefully embraces global developments. The
Bhutanese pride themselves on a sustainable
approach to tourism in line with the philosophy of
Gross National Happiness. Foreign visitors
famously pay a minimum tariff of US$250 per day, making it seem one of the world’s more
expensive destinations. Tourism in Bhutan has diversified from mostly cultural tourists,
sightseers and trekkers to special interests, such as sports and adventure tourism, to ecotourism
and nature-based tourism.
The average length of stay for tourists in Bhutan is 9.6 days. The standard deviation is 2.75
a) What is the probability that a tourist will stay in Bhutan for 7.8 days or fewer?
b) What is the probability that a tourist will stay in Bhutan for 8.9 days or longer?
c) In order to stay longer than 99% of the tourists to Bhutan, how long would a tourist
need to stay?
d) How long would a tourist’s stay have to last in order to be in the shortest 10% of
visits?
Problem #9
budgetyourtrip.com – Bhutan’s strategy of “low volume, high quality”
tourism has made it a highly regarded destination among discerning
travelers. In an effort to protect Bhutan’s environment and culture, the
government has placed a minimum fee of $250 per person per day for
visitors to Bhutan. The amount that includes land transport,
accommodations, food and guide service.
Average daily price for travelling a person in Bhutan is $86 with a
standard deviation of $25.77.
a) What is the probability that people pay more than $140/day?
b) What is the probability that a person pay less than $52/day?

c) What is the probability that an individual pay between $70 and $120/day?

Business Statistics, MGT 233
Chapters 4, 5 & 6 Formula Sheet
Counting Rules
n! n n!
Permutation n PX = – Combination n Cx
(n – x)! x x!(n – x)!
Elementary Probability Rules
P(A u B) = P( A) + P( B) – P(A m B) addition rule
P(A (W B) z P( A) – P(B’A) dependent
P(A m B) = P( A) – P( B) independent
P( A m B)
P(A)B) = – conditional
P(B)
Probability Distribution Rules
Complementary Rule P (X > XI- ) = 1 – P (X S Xi
Distribution Functions
Binomial Probability Distribution f (X = x,- in pk) C X px (1 – [fin-x
E [x] = u = n ‘ p
62 = n – p(1 – p)
j, x

Poisson Probability Distribution f(X = x1. La) 2 -“u
x.
E [x] = u
N ormal Transforms
x – A!
Z Z – X = 1L! + Z O”
O“