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,
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 ra
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
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 , . .
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: Pro
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
Summary: Section 5.2
Summary: Section 5.2
(1) uniform density
f =
1
size()
Summary: Section 5.2
(2) exponential density
X = wait time
= [0, )
=
1
average wait time
f (x) = ex
Summary: Section 5.2
Exa
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
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 = p
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
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 ar
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
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
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 con
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 )
S
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