HW 3 STAT 346, Spring 2013 I'll make each homework assignment ...

HW 3
STAT 346, Spring 2013
I’ll make each homework assignment worth 10 points, so that when I count your best 10 of 12 assignment
scores, your overall homework score will be out of 100 points possible.
Note: Five of the of the eight parts below will be graded, with each graded part worth 2 points.
I’ll instruct the GTA that she can discuss any of the problems from the Spring 2010 HW with you, along
with the Chapter 3 problems on your HW 2 (which are similar to some of these problems) once I post the
solutions to those problems, but that she shouldn’t discuss details of these problems with you.
1) Suppose that a subset of 5 cards will be randomly drawn from an ordinary deck of 52 playing cards. What
is the probability that there will be exactly one black card among the five cards given that there is at least
one black card among the five cards? (Note: Once I post the solutions to HW 2, you might want to consider
an attack similar to what I used in my alternative solution for Probelm 4(b).)
2) In the board game Risk sometimes one player attacks another player with the attacking player rolling
two ordinary six-sided dice and the defending player rolling one ordinary six-sided die. The attacking player
wins only if at least one of his two dice results in more spots on its upward face than what appears on the
upward face of the defender’s die. (E.g., if the attacker rolls a 2 and a 5 and the defender rolls a 4, then the
attacker wins, but if the attacker rolls a 3 and a 4 and the defender rolls a 4, then the defender wins.) What
is the probability that the attacker will win?
3) Consider an urn containing two amber balls, three blue balls, and four green balls. Suppose that one ball
will be randomly removed from the urn and then all balls of the same color as the randomly chosen ball will
also be removed. Finally a ball will will randomly removed from the balls of the two colors that remain in
the urn. (Note: This is the same situation considered in Problem 6 of HW 2. Feel free to use the correct
answer to that problem in your solution for this problem.) What is the probablity that the first ball removed
will be blue given that the last ball removed will be green?
4) Suppose that a person will roll an ordinary six-sided die three times. What is the probability that the
first roll results in one spot on its upward face, the second roll results in two spots on its upward face, and
the third roll results in three spots on its upward face?
5) Suppose that a person will look at the top three cards in a randomly-ordered deck of 52 ordinary playing
cards (consisting of 4 aces, 4 twos, 4 threes, . . ., 4 tens, 4 jacks, 4 queens, and 4 kings). What is the
probability that first card is an ace, the second card is a two, and the third card is a three? (Note: Unlike
the three events considered in Problem 4 above, the events here are not independent.)
6) Consider a roulette wheel having 37 places for the steel marble to land (and suppose that the 37 possibilities
are equally likely). Nine of the places are colored black and numbered with odd integers, nine of the places
are colored black and numbered with even integers, nine of the places are colored red and numbered with
odd integers, nine of the places are colored red and numbered with even integers, and one place is colored
green and numbered with a zero (which is considered to be neither odd nor even when roulette is played).
Letting B be the event that the marble comes to rest in a black place, and E be the event that the marble
comes to rest in an even-numbered place, state whether or not the events B and E are independent and
justify your conclusion quantitatively by showing that the definition of independent events is satisfied or is
not satisfied.
7) Consider Exercise 29 on p. 122 of the text. Ignore the switches located at 5 and 6 (i.e., remove them from
the figure, or equivalently just assume that they are always closed) and give the probability for the case of
p = 0.8.
8) Consider a small school having 50 male students in the 9th and 10th grades, 50 male students in the
11th and 12th grades, and 100 female students (so 200 students in all). Suppose prize 1 will be randomly
given to one of the 50 male students in the 11th and 12th grades, prize 2 will be randomly given to one of
the 100 male students in the school, and prize 3 will be randomly given to one of the 200 students in the
school. Furthermore, suppose that these prizes will be given independently of one another. Suppose that
Wolfgang is a male student in the 11th grade, and is thus eligible for all three of the prizes. Letting X be the

number of these prizes that he will be given, give the pmf of X, expressing all probabilities as exact values
in decimal form. (So don’t round. Each probability should have 6 digits after the decimal, and the sum of
the probabilities for the possible values of X should equal 1 (exactly).)