MATH 407 QUIZ 3 SOLUTIONS
Name:_
Discussion Time:_
1) Suppose that n balls are dropped at random into n bins. What is the probability that exactly one
bin is empty?
(3 pts.)
The number of ways to drop n balls into n bins such that exactly one bin is empty

Math 407, Spring 2012
C. Gupta
Tossing a coin two times. Suppose we toss a coin two times. What is the probability of getting a tail,given that at least one head was observed? The sample space of all possible outcomes is
= cfw_hh, ht, th, tt. The event t

Math 407, Spring 2012
C. Gupta
Observe that R(x) is a decreasing function of x. To see this
R (x) =
n 1
x2
p
1p
< 0.
Therefore, the last time k that R(k )
1 is where Pn,p (k ) takes its maximum value. Let k be the
largest integer for which R(k ) 1 and R(k

Math 407, Spring 2012
C. Gupta
denes a probability function. When = 0 and = 1, we have a special case of this function
12
1
(x) = e 2 x .
2
This special function is called the probability density function of the standard normal distribution. The
probabili

Math 407, Spring 2012
C. Gupta
(iii) Proceed by change of variable. (z ) =
and y (z ) = z. Also, dy = dt. Therefore
z
t2 /2 dt.
1
2 e
z
1
2
ey /2 dy =
2
1 y2 /2
e
dy
2
(z ) =
=
Set y = t. Then y () =
z
z
1
2
ey /2 dy
2
1 y2 /2
e
dy = 1 (z ).
2
A tab

Math 407, Spring 2012
C. Gupta
we assume that it is equally likely that the cholesterol goes down or up. So we assume that
the probability that cholesterol levels go down is 1/2. If X is the number of participants with
lower cholesterol levels at the end

Chapter 3
Random Variables
3.1
Denitions and elementary properties
Denition 3.1.1. Let (, A, P ) be a probability space. A function X with domain and codomain the real number R which satises the condition that for every open interval I , the preimage
X 1

Aug 22.
HW assigned: due Aug 23, but Aug 25 will be tolerated to give people time to buy the text.
Read pages 1-9; ignore page 8. Do exercises 1.1.1-9
Quiz 1 will be on Friday Aug 26. It includes the following:
Q1) What is the largest amount one should ga

Instructor:
Prof. Arratia
Spring 2008
MATH 407
Entropy and information
Twenty questions, L questions. We take a simplied but global view of the game of 20 questions.
I stand in front of the room and write down, in secret, the name of one particular object

Math 407, Spring 2012
2.2
C. Gupta
The Binomial Distribution
Suppose an experiment as a probability p of ending in a favorable outcome, and 1 p of ending
in an unfavorable outcome. Suppose this experiment is repeated n times, independently. What is
the pr

Math 407, Spring 2012
C. Gupta
In this case, since all the balls are distinguishable, we get a total of six (3!) arrangements. But notice that the rst two arrangements correspond to the case RRB , and the two subcases come from
the rearrangement of the re

Math 407, Spring 2012
1.5
C. Gupta
Independence
We say that events A and B are independent if the information that B has occurred gives us no
information about the probability that A has occurred. In terms of conditional probability, this
means that
P (A|

Math 407, Spring 2012
1.6
C. Gupta
Bayes Rule
Theorem 1.6.1. Suppose B1 , B2 , . . . , Bn is a partition of . Let A be an event with P (A) > 0.
Then
P (A|Bi )P (Bi )
P (Bi |A) = n
.
j =1 P (A|Bj )P (Bj )
Proof.
P (Bi |A) = P (Bi A)/P (A) = P (A|Bi )P (Bi

Math 407, Spring 2012
1.7
C. Gupta
Some useful formulae and inequalities
Theorem 1.7.1 (Inclusion-Exclusion Rule for 3 events).
P (A B C ) = P (A) + P (B ) + P (C ) P (A B ) P (B C ) P (C A) + P (A B C )
Proof. We know that P (AB ) = P (A)+P (B )P (AB ).

Math 407, Spring 2012
1.8
C. Gupta
Sequences of events
Suppose we have a sequence of events A1 , A2 , . . . , An . Then
P (A1 A2 . . . An ) =
=
.
.
.
=
P (An |An1 A1 )P (An1 A1 )
P (An |An1 A1 )P (An1 |An2 A1 )P (An2 . . . A1 )
.
.
.
P (An |An1 A1 )P (An1

Math 407, Spring 2012
C. Gupta
Let us compute the probability that he wins 0 races of 1 race. Since the outcomes of the races
are independent, he wins 0 with probability r0 = (1 p1 )(1 p2 )(1 p3 )(1 p4 ). If we let Bi
denote the event that he wins the ith

Math 407, Spring 2012
C. Gupta
the employee is single, the company provides 1 pension. If the employee is still married the
company provides 1.5 pensions.
Past information shows that of the people who retire (complete 20 years) with the company,
the fract

Math 407, Spring 2012
C. Gupta
The birthday problem. Suppose there are n students in a class. What is the probability that at
least two students in the class share a birthday?
To compute the probability of this event, we need to be able to establish when

Chapter 2
Repeated Trials
2.1
Counting
Suppose we have n distinguishable objects: say numbered balls. We want to know in how many
ways these balls can be lined up on a table. Each line is uniquely determined by the sequence in
which the numbers on the bal

Instructor:
Prof. Arratia
MATH 505b
HW due Feb 1, maximal coupling and dT V
Jan 25, 2012
Note that the text, on page 128 (7) gives the analysts version of total variation distance,
which is twice the distance that we are using in class.
The third from bot

Instructor:
Prof. Arratia
MAIH 4 07
Midterm 1
September 0
3
8o
SHOW Y OUR W ORK
1.) [ 30:5each]
P(A) : . 4,P (B) : . 5,F (C) : . 6
F(A n B ) : . 2L,P (A n C ) : . 32,P (B n C ) : . JI,
P(An BnC):.16.
a) Draw the Venn diagram ("opy from the board) and fill

Instructor:
Prof. Arratia
MATH 4 07
Midterm 2
November i, 2 011
1
:r't
SHOW Y OUR W ORK
p
t.) 1 27 oints t otall
14,.)[3:1+2] I n t he t rue s entence The C LT a ppliest o s umsa nd averagesof a large number
"
of i id r andomv ariables."
i
C L T i s a n a

PROBLEM OF THE WEEK 2
A and B take turns rolling two dices. The rst one to get a sum larger or equal than 10 wins.
- What is the probability of having a sum larger than 10 (at the rst roll)?
- Find the probability that the game lasts exactly k turns.
- If

PROBLEM OF THE WEEK 3
Here is the game: youre ipping coins turn after turn, an innite number of time. At
each turn, you get a new coin: for the nth turn, the probability of having H (for Head) is
pn , and the probability of having T (for Tail) is 1 pn . A

PROBLEM OF THE WEEK 4
Preliminaries:
(0) Show that for any random variable X with values in N, one has
P (X n).
E [X ] =
nN
Problem:
You are waiting for the bus, but there are two buses that bring you home. Let us denote
T1 the time it takes bus 1 to arri

Problem of the Week 5
Someone gave you a biased coin, but you dont know how biased it is. In other words, the
parameter p = P (Heads) is unknown, and we want to estimate it.
(1)
(2)
(3)
Let X be the number of Heads that appear performing n independent coi

MATH 407 QUIZ 1 SOLUTIONS
1) For m, n, k , how many distinct integer-valued vectors ( x1 ,., xk ) satisfy x1 + . + xk = n and
x1 ,., xk > m ? Justify your answer. (3 pts.)
We saw in discussion that the number of positive integer-valued vectors ( x1 ,., xk

MATH 407 QUIZ 2 SOLUTIONS
1) In certain variants of poker, a hand is called a flush house if it consists of 3 cards of one suit and 2
cards of another. Given a perfectly shuffled deck of cards, what is the probability of being dealt a flush
house? You do

MATH 407 QUIZ 4 SOLUTIONS
Name:_
Discussion Time:_
1) Let X be a binomial random variable with parameters n and p (0,1) - that is, X has
n +1
n
1 1 (1 p )
.
probability mass function f X (k ) = p k (1 p) n k , k = 0,1,., n. Show that E
=
p (n + 1)
1 + X

August 23, 2013
Probability Theory: Homework problems
Homework 1.
1. A traditional three-digit telephone area code is constructed as follows. The rst
digit is from the set cfw_2, 3, 4, 5, 6, 7, 8, 9, the second is either 0 or 1, the last is from the set
c