STAT/MA 41600
Practice Problems: September 5, 2014
1. Choosing a page at random.
A student buys a brand new calculus textbook that has 1000 pages, each numbered with
3 digits, from 000 to 999. She randomly opens the book to a page and starts to read! Assu
STAT/MA 41600
In-Class Problem Set #9: September 15, 2014
Solutions by Mark Daniel Ward
1a. The joint mass is pX,Y (1, 1) = (3/5)(2/4) = 3/10, pX,Y (1, 0) = (3/5)(2/4) = 3/10,
pX,Y (0, 1) = (2/5)(3/4) = 3/10, and pX,Y (0, 0) = (2/5)(1/4) = 1/10.
1b. We ha
STAT/MA 41600
In-Class Problem Set #9: September 15, 2014
1. Suppose Alice and Bob each take a cookie, without replacement, from a jar that contains
5 cookies, 3 of which are chocolate, and the other 2 are non-chocolate. Let X = 1 if Alice
gets chocolate;
STAT/MA 41600
Practice Problems: September 15, 2014
Solutions by Mark Daniel Ward
1. Butteries. Alice, Bob, and Charlotte are looking for butteries. They look in three
separate parts of a eld, so that their probabilities of success do not aect each other.
STAT/MA 41600
Practice Problems: September 15, 2014
1. Butteries. Alice, Bob, and Charlotte are looking for butteries. They look in three
separate parts of a eld, so that their probabilities of success do not aect each other.
Alice nds 1 buttery with pro
STAT/MA 41600
In-Class Problem Set #8: September 12, 2014
Solutions by Mark Daniel Ward
1. The mass of X is pX (x) = (32/52)x1 (20/52), for integers x 1. So the CDF of X, for
x
an integer x 1, is FX (x) = x (32/52)j1 (20/52) = (20/52) 1(32/52) = 1 (32/52)
STAT/MA 41600
In-Class Problem Set #8: September 12, 2014
1. Suppose that we choose cards from a standard 52-card deck, with replacement and
shuing in between cards, until the rst card with value 6, 7, 8, 9, or 10 appears, and then
we stop. Let X be the n
STAT/MA 41600
Practice Problems: September 12, 2014
Solutions by Mark Daniel Ward
1. Butteries. Alice, Bob, and Charlotte are looking for butteries. They look in three
separate parts of a eld, so that their probabilities of success do not aect each other.
STAT/MA 41600
Practice Problems: September 12, 2014
1. Butteries. Alice, Bob, and Charlotte are looking for butteries. They look in three
separate parts of a eld, so that their probabilities of success do not aect each other.
Alice nds 1 buttery with pro
STAT/MA 41600
In-Class Problem Set #7: September 10, 2014
Solutions by Mark Daniel Ward
1. There are 63 = 216 equally-likely possible outcomes. So (1a) we have P (X = 0) =
3
125
25
(1/6)0 (5/6)3 = 216 ; (1b) we have P (X = 1) = 3 (1/6)1 (5/6)2 = 72 ; (1c)
STAT/MA 41600
In-Class Problem Set #7: September 10, 2014
(there is no Problem Set #6)
1. Roll three (6-sided) dice. Let X denote the number of 2s that appear.
1a. Find P (X = 0). 1b. Find P (X = 1). 1c. Find P (X = 2). 1d. Find P (X = 3).
2. Suppose that
STAT/MA 41600
Practice Problems: September 10, 2014
Solutions by Mark Daniel Ward
1. Harmonicas. Since X is a waiting time, then X takes value in the interval [0, ), so X
is a continuous random variable.
Since Y is a nonnegative integer, i.e., Y takes val
STAT/MA 41600
Practice Problems: September 10, 2014
1. Harmonicas. When ordering a new box of harmonicas, let X denote the time until the
box arrives, and let Y denote the number of harmonicas that work properly.
Is X a continuous or discrete random varia
STAT/MA 41600
In-Class Problem Set #5: September 8, 2014
Solutions by Mark Daniel Ward
1. Let A be the event that the random student lives in a residence hall, and let B be the
P (B|A)P (A)
(AB)
event that the student arrived on-time. Then P (A|B) = P P (
STAT/MA 41600
In-Class Problem Set #5: September 8, 2014
1. At a certain college, 40% of the students live in a residence hall (on-campus), and the
other 60% of the students live o-campus. Suppose that students who live in the residence
hall arrive to cla
STAT/MA 41600
Practice Problems: September 8, 2014
Solutions by Mark Daniel Ward
1. Waking up at random. 1a. Writing A as the event it is a weekday, and B as the event
it is before 8 AM, we have
P (A | B) =
P (B | A)P (A)
(.65)(5/7)
P (A B)
=
=
= .881.
c
STAT/MA 41600
Practice Problems: September 8, 2014
1. Waking up at random.
On each weekday, a student wakes up before 8 AM with probability .65, or after 8 AM
with probability .35.
On each weekend, a student wakes up before 8 AM with probability .22, or a
STAT/MA 41600
In-Class Problem Set #4: September 5, 2014
Solutions by Mark Daniel Ward
1. Let A denote the event that the two results are equal; let B denote the event that the
result on the red die is less than or equal to the result on the green die. Th
STAT/MA 41600
In-Class Problem Set #4: September 5, 2014
1. Roll a red die and a green die. Given that the result on the red die is less than or equal
to the result on the green die, nd the probability that the two results are equal.
2. Suppose that a dra
STAT/MA 41600
Practice Problems: September 5, 2014
Solutions by Mark Daniel Ward
(AB)
1. Choosing a page at random. (a.) We have P (A | B) = P P (B) . Since A B, then
P (A B) = P (A) = 1/1000. Also P (B) = 271/1000, as discovered on problem set 2. Thus
1/
STAT/MA 41600
In-Class Problem Set #3: September 3, 2014
Solutions by Mark Daniel Ward
1. Events A and B are independent. Why? We note P (A) = 4/8 = 1/2 and P (B) = 4/8 =
1/2, and P (A B) = P (cfw_c, d) = 2/8 = 1/4, so P (A)P (B) = (1/2)(1/2) = 1/4 = P (A
STAT/MA 41600
In-Class Problem Set #3: September 3, 2014
1. Consider a sample space S with eight outcomes, S = cfw_a, b, c, d, e, f, g, h. Suppose
that each outcome is equally likely to appear. Now dene the events A = cfw_a, b, c, d and
B = cfw_c, d, e, f
STAT/MA 41600
Practice Problems: September 3, 2014
Solutions by Mark Daniel Ward
1. Choosing a page at random. Yes, the events are independent. We compute
P (A) = 100/1000 = 1/10;
P (B) = 10/1000 = 1/100;
P (A B) = 1/1000.
So P (A)P (B) = P (A B). So A, B
STAT/MA 41600
Practice Problems: September 3, 2014
1. Choosing a page at random.
A student buys a brand new calculus textbook that has 1000 pages, each numbered with
3 digits, from 000 to 999. She randomly opens the book to a page and starts to read! Assu
STAT/MA 41600
In-Class Problem Set #1: August 27, 2014
Solutions by Mark Daniel Ward
Problem Set 1 Answers
1a. When building an event that contains outcome a, we have 6 choices to make, each with
two possibilities: Should b be included, or not? Should c b
STAT/MA 41600
In-Class Problem Set #1: August 27, 2014
1. Consider a sample space S with seven outcomes, e.g., S = cfw_a, b, c, d, e, f, g.
(a.) How many events are there, which contain outcome a?
(b.) How many events are there, which contain outcome a an
STAT/MA 41600
Practice Problems: August 27, 2014
Solutions by Mark Daniel Ward
Problem Set 1 Answers
1. Choosing points at random. (a.) The sample space is
S = cfw_(x, y) | 0 x 2, 0 y 2 x.
(b.) The sample space is
S = cfw_(x, y) | 0 x 2, 0 y 4 x.
2. Glove
STAT/MA 41600
Practice Problems: August 27, 2014
1. Choosing points at random.
(a.) A point is chosen at random inside in the triangle in Figure 1. What is the sample
space? [Please give mathematical expression(s), rather than just (x, y) is in the triang
STAT/MA 41600
In-Class Problem Set #2: August 29, 2014
Solutions by Mark Daniel Ward
1a. Let Aj denote the event that the jth die has value 2. Then P (A1 A2 A3 ) =
P (A1 ) + P (A2 ) + P (A3 ) P (A1 A2 ) P (A1 A3 ) P (A2 A3 ) + P (A1 A2 A3 ) =
1/6+1/6+1/6(
STAT/MA 41600
In-Class Problem Set #2: August 29, 2014
1. Roll three (6-sided) dice.
1a. Use inclusion-exclusion to nd the probability that at least one value of 2 appears.
1b. Find the probability that at most one value of 2 appears.
2. Is it always true