Charles M. Grinstead and J. Laurie Snell: INTRODUCTION to PROBABILITY Published by AMS Solutions to the exercises SECTION 1.1 1. 3. 5. 7. 9. As n increases, the proportion of heads gets closer to 1/2, but the difference between the number of heads and hal
Welcome to Math 205: Probability!
This course analyzes repeatable experiments, such as coin tosses or die rolls, in which shortterm
outcomes are uncertain, but longrun behavior is predictable. Such random processes are used as
models for realworld phenome
Summary: Chapter 11
Summary: Chapter 11
A Markov chain consists of a set of states
S = cfw_s1 , s2 , . . . , sr
together with a transition matrix
p11 p12 . . .
p21 p22 . . .
P= .
.
.
.
.
.
.
.
pr 1 pr 2 . . .
p1r
p2r
.
.
.
,
prr
where pij is the transi
Useful Facts:
1. P(E) 0 for every E .
2. P() = 1.
3. If E F , then P(E) P(F ).
4. If A and B are disjoint subsets of ,
then P(A B) = P(A) + P(B).
5. If P(A) = 1 P(A) for every A .
6. If A1 , A2 , . . . , An are pairwise disjoint subsets of , then
n
P(A1 A
Solutions: Problems for Chapter 3
Solutions: Problems for Chapter 3
Problem A:
You are dealt ve cards from a standard deck. Are you more
likely to be dealt two pairs or three of a kind?
Solutions: Problems for Chapter 3
experiment: choose 5 cards at rando
Summary: Sections 6.1 and 6.2, Part 2
Summary: Sections 6.1 and 6.2, Part 2
X discrete, numerical random variable
sample space
m probability distribution function
Summary: Sections 6.1 and 6.2, Part 2
The mean or expected value of X is
or E(X ) =
x m(x)
Summary: Sections 6.1 and 6.2, Part 1
Summary: Sections 6.1 and 6.2, Part 1
X discrete, numerical random variable
sample space
m probability distribution function
Summary: Sections 6.1 and 6.2, Part 1
The mean or expected value of X is
or E(X ) =
x m(x)
Summary: Section 5.1
Summary: Section 5.1
(1) uniform distribution
= cfw_1 , 2 , . . . , n
m() =
1
n
Summary: Section 5.1
(2) binomial distribution
X = number of successes
n = number of trials
p = probability of success in one trial; q = 1 p
= cfw_0, 1
Summary: Chapter 4
Summary: Chapter 4
P(F |E) denotes the conditional probability that the event F
occurs given that the event E has occurred.
P(F |E) =
P(F E)
P(E)
Summary: Chapter 4
Two events E and F are independent if P(F |E) = P(F ) and
P(E|F ) = P(E
Summary: Chapter 3
Summary: Chapter 3
Multiplication Principle:
A task is to be carried out in a sequence of r stages. There are
n1 ways to carry out the rst stage; for each of these n1 ways,
there are n2 ways to carry out the second stage; for each of
th
Introduction: Chapter 5
Introduction: Chapter 5
The goal of probability theory is to assign probabilities
(numerical measures of likeliness) to events in useful ways
Introduction: Chapter 5
We do this for a given experiment by creating a probability
model
Example 2.14
Example 2.14
X = sum of two random numbers from [0, 1]
= [0, 2]
Find the cumulative distribution function F (x), and determine
the probability density function f (x).
Compute P
1
2
X
3
2
.
Example 2.14
U = random number from [0, 1]
V = rand
Summary: Section 2.2
Summary: Section 2.2
distribution function for discrete X :
m:R
1. m() 0 for all
2.
m() = 1
probability of an event E:
P(E) =
m().
E
Summary: Section 2.2
density function for continuous X :
f :R
1. f (x) 0 for all x
2.
f (x)dx = 1
p
Summary: Section 1.2
Summary: Section 1.2
random variable: X (denotes the outcome of an experiment)
sample space: (the set of all possible values for X )
outcome: (element of )
event: E (subset of )
Summary: Section 1.2
X is discrete is nite or countably
Math 205: Probability
Math 205: Probability
What is probability?
Math 205: Probability
A probability is a number that we assign to a given event to
describe its likeliness. The larger the number, the more likely
the event.
Math 205: Probability
Example:
W