Probability and Statistics
MATH2010
November 17, 2023
Midterm Mocktest
Problem 1
Qingque is a Celestial Jade addict and she always bring a board and a bunch of jade
tiles together with her to play during work hours. In this game, each turn, Qingque
will draw 2 jade tiles from 3 different suits and can hold up to 4 tiles at one time on
the board. If the number of tiles on the board is already 4, Qingque will choose the
combination of 4 tiles (out of 6) so that there are the most number of tiles with the
same suit on the board. In order to win, she needs to achive the “Hidden Hand” state,
which is 4 jade tiles on the board are from the same suit.
1. Given that there are 3 tiles from a same suit on the board. Find the probability
of Qingque achieving the “Hidden Hand” state in the next turn
2. Calculate the probablility of Qingque achieving the “Hidden Hand” state in less
than 4 turn
Problem 2
A group of shooters consists of 3 experts and 4 intermediates . The probabilities of
hitting the target for each type are 0.9 and 0.8, respectively. Two shooters are randomly selected, and each person takes one shot. Given that at least 1 person misses
the target in the 2 shots, calculate the probability that both individuals are experts.
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MATH2010
November 17, 2023
Probability and Statistics
Problem 3
Three employees from different departments of a company are being considered for a ”Best Employee of the Year” award. Their performance ratings are given on different scales. Determine which employee has performed best relative to their department’s standards.
Employee Performance Rating
Department Average
Department Rating
Rating
Standard Deviation
Emily
Raj
Carlos
78
4.2
88
70
3.5
80
10
0.5
20
Problem 4
Several food delivery apps claim to offer a wide variety of restaurant options and fast
delivery times. A crucial factor for customer satisfaction is the actual diversity of
cuisines and the accuracy of the delivery times promised. If the app does not meet
these expectations, customers may look elsewhere.
A local food blogger aims to analyze the performance of these food delivery apps. She
selects one restaurant from each of the following cuisine categories: Italian, Chinese,
Indian, Mexican, and Thai, and orders the most popular dish from each through five
different apps. She evaluates the diversity of the options available and the accuracy of
the delivery times, subsequently writing a blog post about her findings.
Write an analysis of her study that addresses the following issues: Is her sample representative of the population of all college textbooks? Explain why or why not. Describe
some possible sources of bias in this study, and how it might affect the results of the
study. Give some suggestions about what could be done to improve the study.
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Probability and Statistics
MATH2010
November 17, 2023
Problem 5
Harry is a (relatively untalented) basketball player practicing free throws. Each time
Harry attempts a shot, Harry has a p probability of making the shot (independently
of all other shots taken). Let X be the number of shots that Harry makes after 100
attempts.
For question 1 to 4, let p = 0.01.
1. What type of discrete PMF is X? Compute the variance Var[X]. (Given an exact answer, not an approximation.)
2. Use an appropriate approximation to give an estimate for the probability that
X = 2.
3. Harry is hoping the event X > 1 will occur, since this would mean that (thanks
to good luck) Harry made more shots than he expected to make (which might
impress any talent scouts who happened to present). Use the same approximation in question 2 to estimate P {X > 1}.
4. Using the same approximation in question 2, estimate P {X < 1}. Is this larger
or smaller than (or the same as) your estimate for P {X > 1}? Interpret the
comparison result.
For question 5 to 6, let p = 0.1225.
5. Use an appropriate approximation to give an estimate for the probability that
X ≤ 15.
6. Harry is betting that he will make more than k shots. He will win 2 dollars if he
is correct, otherwise he will lose 1 dollar. Find maximum k so that his expected
earning is positive. Make use of the approximation in question 5.
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Probability and Statistics
MATH2010
November 17, 2023
Problem 6
One often hears that the death rate of a person who smokes is, at each age, twice that
of a nonsmoker. What does this mean? Does it mean that a nonsmoker has twice the
probability of surviving a given number of years as does a smoker of the same age?
In an explicit way, denote λs (t) as the hazard rate of a smoker of age t and λn (t) as
that of a non-smoker of age t. Then the statement at issue is equivalent to the statement that
λs (t) = 2λn (t)
1. Prove that
P (ts > A|ts > B) = P (tn > A|tn > B)2
where ts , tn be the age of smoker, non-smoker, and then with a > b
P (ts > A|ts > B) be the probability that a B-year-old smoker will survive
until age A,
P (tn > A|tn > B) be the probability that a B-year-old nonsmoker will survive
until age A.
2. Given the hazard rate of a smoker of age t, with assumption that he has a lung
cancer, such that
λs (t) = 0.027 + 0.00025(t − 40)2 , t ≥ 40
Assuming that a 40-year-old male smoker survives all other hazards, what is the
probability that he survives to age 60 without contracting lung cancer?
Using the result from part 1, and the assumption that the death rate of a person
who smokes is, at each age, twice that of a nonsmoker, calculate the probability
that a non-smoker survives to age 60 without contracting lung cancer.
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