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Salins MA 581 Midterm Exam Feb 27, 2019
Instructions:
• Write all of your answers in the blue book.
• Explain all of your steps.
• There is no need to simplify arithmetic (unless stated otherwise). Leave
factorials, combinations, and sums in your answer.
• You may not use notes, books, calculators, phones, or the internet.
• Do not cheat.
• Some summation formulas are on the back page.
1. (20 points)Answer the following combinatorial probability questions.
(a) A bucket contains 35 red balls, 45 blue balls, and 10 orange balls.
You select 10 balls out of the bucket at random without replacement.
What is the probability of selecting exactly 5 red balls, 4
blue balls, and 1 orange ball? Explain all of your steps and do not
simplify your arithmetic.
(b) Roll 6 fair dice. A fair die has a 1
6
probability of landing on each of
the possible six outcomes 1,2,3,4,5, and 6. Each die is independent
of all of the others. What is the probability that none of the dice
match each other? Explain all of your steps and do not simplify
your arithmetic.
Salins MA 581 Midterm Exam Feb 27, 2019
2. (30 points) A hiring director at a company has found that only 5% of
people who see a particular job listing are qualified for the advertised
position. Additionally, she found that 70% of the qualified people who
see the job listing choose to apply for the position while 1% of the unqualified
people who see the job listing choose to apply for the position.
Let A be the event that a person who saw the job listing applies for
the job. Let Q be the event that a person who sees the job listing is
qualified for the job.
(a) Write each of these sentences in terms of A, Q, probabilities, conditional
probabilities, equal signs, and/or numbers (Your answer
to each part should look something like “P(A|Q) = .7”). You do
not need to show work or explain your answers for this part.
i. “5% of people who see a particular job listing are qualified for
the advertised position.”
ii. “70% of the qualified people who see the job listing choose to
apply for the position.”
iii. “1% of the unqualified people who see the job listing choose
to apply for the position.”
(b) Given that a person who saw the job listing applied for the position,
what is the probability that this person is qualified?
Explain all of your steps and write your work in a clear and organized
manner. There is no need to simplify your arithmetic.
Salins MA 581 Midterm Exam Feb 27, 2019
3. (30 points) You roll a fair die 10 times. A fair die means that the
probability of each outcome 1,2,3,4,5, or 6 is 1
6
. Each roll of the die
is independent of all of the other rolls. Let X be a random variable
that represents the number of times the die lands on 2 during these ten
rolls.
(a) What are the possible values of X? Explain your answer.
(b) Write down the pmf for X. Explain your answer.
(c) Calculate P(X ≤ 4). Show all of your work. There is no need to
simplify arithmetic.
(d) Calculate the probability that X is an even number. Show all of
your work. There is no need to simplify arithmetic.
4. (20 points) A random variable X has the pmf
0 otherwise
where C is some constant that you need to solve for.
(a) Find the value of C that makes pX a valid pmf. Show that your
choice satisfies all of the requirements of a pmf. For this problem
you should actually simplify your arithmetic.
(b) Calculate P(X ≤ 4). Show your work. Simplify your arithmetic.
(c) Calculate P(X ≤ 2|X ≤ 4). Show your work. There is no need to
simplify your arithmetic.
Salins MA 581 Midterm Exam Feb 27, 2019
Summation formulas
• Binomial Theorem.
• Exponential Taylor series.
• Sine Taylor series

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