Homework 2.Homework 3. Homework 4. Homework 5 Homework 6 Homework 7.
Homework set 1 (due January 16):
1. Prove the following statements about any probability function
P and any sets A and B.
(a) P(A ∪ B) ≤ P(A) + P(B)
(b) P(A ∩ B) ≤ max(P(A), P(B))
(c) P(A ∩ B) ≤ P(A) + P(B)
2. A tourist wants to visit six of America's ten largest cities.
In how many ways can she do that if the order of her visits
(a) matters to her?
(b) does not matter?
3. Let A be the event that a person is male, B
that the person is under 30, and C that the person speaks
French. Describe in set theory symbols (a Venn diagram may
help)
(a) a male at least 30 years old
(b) a female under 30 who speaks French
(c) a male who either is under 30 or speaks French
4. Suppose n people are seated at random in a row
of n seats. How many of the n! possible
combinations of people have Ms. Jones and Ms. Smith sitting next
to each other?
5. By considering the position of the leftmost object selected, show that
Practice set 1 (You do not hand these
in.) (Solution.)
Book problems 1.1.4, 1.1.7, 1.1.8, 1.1.9, 2.1.6
Homework set 2 (due January 23)
1. How many partial derivatives of order 3 are there for a
function of three variables? (Recall that for a partial
derivative it only matters how many times you differentiate with
respect to each variable, not the order in which you do it)
2. 12 fair coins are flipped. What is the chance of
getting more heads than tails?
3. When are the following true?
(a) A∪(B∩C)=(A∪B)∩(A∪C)
(b) A∪(B∪C)=A\(B\C) (here D\E stands for D∩Ec
(c) A\(B∪C)=(A\B)∩(A\C)
4. If all poker hands (five cards out of 52) are equally likely,
what is the probability of a flush (meaning that all five cards
are of the same suit)?
5. Given two events, A and B, with P(A)=0.5 and P(B)=0.8, what are the smallest and largest possible values for P(A ∩ B)?
Practice set 2 (Solution)
Book problems 2.1.7, 2.1.8, 2.1.9, 2.1.12, 2.1.16
Homework set 3 (due
January 30)
1. Show that P(AΔB) = P(A) + P(B) - 2 P(A ∩ B)
where AΔB=(A\B)∪(B\A) is the event that exactly
one of A and B happens.
2. 75% of US students graduate from high school. 72% of those go to college, and 56% of those graduate with a Bachelor's degree (within six years). What percent of students starting elementary school get a Bachelor's degree?
3. Suppose we roll two dice twice. What is the probability that we get the same sum both times?Homework 4 (due February 6)
1. A group of families have 1, 2 or 3 children with probability
1/3 each.
(a) Robert has no brothers. What is the probability he is an
only child.
(b) Rupert has no sisters. What is the probability he is an only
child.
2. A particular football team is known to run 40% of its plays to the left and 60% to the right. When the play goes to the right, the right tackle shifts his stance 80% of the time, but does so only 10% of the time when the play goes to the left. As the team lines up for the next play, the right tackle shifts his stance. What is the chance that the play will go to the right?
3. Charlie draws five cards out of a deck of 52. If he gets at least three cards of one suit, he discards the cards not of that suit, and draws as many cards as he discarded. What is the probability he ends up with five cards of the same suit?
4. Five pennies are sitting on a table. One is a trick coin with heads on both sides, while the other four are normal. You pick a penny at random, flip it four times (without inspecting it) and get four heads. Do you believe that you got the two-heads penny? (Your argument should be based on a probability)
5. Show that P(A|B) > P(A) if and only if P(B|Ac) < P(B|A).
Practice set 4 (solutions
)
Book problems 2.3.1, 2.3.2, 2.3.4, 2.3.7, 2.3.8 (including the
small print), 2.3.11.
Homework 5 (due February 13)
1. Let A and B be two events with P(B) > 0. Show that
(a) if B ⊆ A then P(A|B) = 1
(b) If A ⊆ B then P(A|B) = P(A)/P(B).
2. (a) An event A is independent of itself. What is its
probability?
(b) If A is independent of B, is A independent of Bc?
3. You roll a die once. If it shows j, you roll it j-1 times
more, and add all the j scores. If this sum is 3, what is the
probability that
(a) you rolled the die once altogether?
(b) you rolled the die twice altogehter?
4. Two golfers are playing sudden death to decide a tournament. The first one wins a hole with probability p, the second one wins with probability q, and holes are tied with probability r. Holes are independent, and the game stops the first time someone wins a hole. What is the probability that the first player wins?
5. Two roads join Ashville and Benson, and two further roads join Benson to Carlyle. Ashville is directly connected to Carlyle using a railroad. All four roads and the railroad are independently blocked by mudslides with probability p. If you are at Ashville, what is the probability you can travel to Carlyle, and if you get there, what is the probability that the railroad is blocked?
Practice set 5 (solutions )
Book problems 2.4.1, 2.4.4, 2.4.7, 2.4.12.
Homework 6 (due February 28) Rewrite completely any midterm problem that you lost points on. Please hand in your exam paper as well, so we can see which problems are relevant. You may recover up to half of your missed points.
Practice set 6 (Solutions)
Book problems 1.2.1, 1.2.10, 1.2.14, 3.1.7
Homework 7 (due March 7)
1. A family has n > 0 children with probability αpn,
where α≤p/(1-p).
(a) What is the probability that the family has no children?
(b) If each child is equally likely to be a boy or a girl
(independently of each other), what proportion of families
consists of k boys and any number of girls?
2. Let X have pdf 2 exp(-2x) for x>0. What is the pdf of Y=1-exp(-2X)?
3. Suppose a particle moves along the x-axis, starting at 0. It
moves a step to the right or to the left with equal probability
(you may assume that all steps are of unit length).
(a) What is the pmf of its position after 4 steps?
(b) What would the pmf be if it is twice as likely to go to the
right as to the left?
4. Let n be a positive integer. Show that fX(x)=(n+2)(n+1)xn(1-x), 0≤x≤1, is a pdf.
5. Suppose that fX(x) is a continuous pdf, with the symmetry property that fX(a)=fX(-a) for any a>0. Show that P(-a ≤ X ≤ a) = 2 FX (a) -1, where FX(x) is the cdf corresponding to the pdf fX(x).
Practice set 7 (Solutions)
Book problems 3.4.3, 3.4.11, 3.5.5, 3.5.8, 3.5.10, 3.5.12