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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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