2021/03/18
Statistics 1A
Short History of Probability
French Society in the
1650’s:
• Gambling was popular
and fashionable
• Not restricted by law
• As the games became
more complicated and
the stakes became
larger there was a
need for mathematical
methods for computing
chances
Short History of Probability
Enter the Mathematicians:
• A well-known gambler, the
chevalier De Méré consulted
Blaise Pascal in Paris about a
some questions about some
games of chance
• Pascal began to correspond
with his friend Pierre Fermat
about these problems
1
2021/03/18
Short History of Probability
• The correspondence between
Pascal and Fermat is the
origin of the mathematical
study of probability
• The method they developed is
now called the classical
approach to computing
probabilities
Short History of Probability
It is better to be satisfied with
probabilities than to demand
impossibilities and starve.
Johann Christoph Friedrich von Schiller (1759 - 1805)
What is Probability?
Imagine that you were living in the
seventeenth century as a nobleman. One day
your friend Chevalier de Méré was visiting
and challenged you to a game of chance.
You agreed to play the game with him.
He said, "I can get a sum of 8 and a sum of 6
rolling two dice before you can get two sums
of 7’s."
Would you continue to play the game?
2
2021/03/18
What is Probability?
I hope you changed your mind about playing
the game of chance!
Chevalier de Méré did not state the order for
the 8 and the 6.
11
21
31
41
51
61
12
22
32
42
52
62
13
23
33
43
53
63
14
24
34
44
54
64
15
25
35
45
55
65
16
26
36
46
56
66
What is Probability?
So, there are ten possible ways for him to get
a favourable result.
There are only six ways for you to get a
favourable result.
I hope you imagined yourself as a wealthy
nobleman, because you have a 54.6 percent
chance of losing if you accepted to play the
game with Chevalier de Méré.
Sample Spaces and Events
An experiment is any action or process whose
outcome is subject to uncertainty:
•
•
•
•
Record an age
Toss a die
Record an opinion (yes, no)
Test a fuse to determine whether it is
defective
9
3
2021/03/18
Sample Spaces and Events
Definition:
The sample space of an experiment, denoted by
S, is the set of all possible outcomes of that
experiment
Example 2.1
Example 2.2
Example 2.3
Example 2.4
10
Sample Spaces and Events
Definition:
An event is any collection (subset) of outcomes
contained in the sample space S.
An event is said to be simple if it consists of
exactly one outcome and compound
if it consists of more than one outcome.
Example 2.5
Example 2.6
Example 2.7
11
Relations from Set Theory
An event is nothing but a set, so relationships and results
from elementary set theory can be used to study events.
Definition:
1. The union of two events A and B, denoted by A B and
read “A or B,” is the event consisting of all outcomes
that are either in A or in B or in both events - that is,
all outcomes in at least one of the events.
A B
S
A
B
12
4
2021/03/18
Relations from Set Theory
Definition:
2. The intersection of two events A and B, denoted by
A  B and read “A and B,” is the event consisting of all
outcomes that are in both A and B
A B
S
A
B
13
Relations from Set Theory
Definition:
3. The complement of an event A, denoted by A ' is the
set of all outcomes in S that are NOT contained in A .
A'
S
A
B
14
Relations from Set Theory
Definition:
When A and B have no outcomes in common, they are said
to be disjoint or mutually exclusive events.
It is written as
Example 2.8
Example 2.9
Example 2.10
A B  
Null or
empty
event
15
5
2021/03/18
Axioms and Properties of Probabilities
Given an experiment and a sample space S, the objective
of probability is to assign to each event A a number P(A),
called the probability of the event A, which will give a
precise measure of the chance that A will occur.
To ensure that the probability assignments will be
consistent with our intuitive notions of probability, all
assignments should satisfy the following axioms (basic
properties) of probability.
16
Axioms and Properties of Probabilities
Let S denote the sample space of an experiment.
The probability of an event A in S, denoted by P(A), is a
number for which the following conditions hold:
1.
P  A  0
2.
PS   1
3. For an infinite collection of mutually exclusive
events A , A ,...  S
1
2
it follows that

P A1  A2  ...   P Ai 
17
i 1
Propositions
1. P    0
Null event
2. For any event A, P A  1  P A '
Prove!
3. For any events A and B,
P  A  B   P  A  P B   P  A  B 
Prove!
Example 2.14
18
6
2021/03/18
Determining Probabilities
• A simple event is the outcome that is
observed on a single repetition of the
experiment
• The basic element to which probability is applied
• One and only one simple event can occur when the
experiment is performed
• A simple event is denoted by E with a
subscript
Determining Probabilities
• Each simple event will be assigned a
probability, measuring “how often” it occurs
• The set of all simple events of an experiment
forms the sample space, S
Example
• The die toss:
• Simple events:
1
E1
2
E2
3
E3
4
E4
5
E5
6
E6
Sample space:
S ={E1, E2, E3, E4, E5, E6}
•E1
S
•E3
•E5
•E2
•E4
•E6
7
2021/03/18
Example
• Toss a fair coin twice. What is the probability
of observing at least one head?
Axiom 3
1st Coin
2nd Coin Ei
H
HH
P(Ei)
Axiom 1
1/4
P(at least 1 head) =
T
HT
1/4
P(HH) + P(HT) + P(TH)
H
TH
1/4
= P(E1) + P(E2) + P(E3)
T
TT
1/4
= 1/4 + 1/4 + 1/4 = 3/4
H
T
1
Axiom 2
Example
• A bowl contains three M&Ms®, one red, one
blue and one green. A child selects two M&Ms
at random. What is the probability that at
least one is red?
1st M&M 2nd M&M
m
m
m
m
m
RB
P(Ei)
1/6
RG
1/6
P(at least 1 red)
BR
1/6
BG
1/6
= P(RB) + P(BR)+ P(RG)
+ P(GR)
m
GB
1/6
= 4/6 = 2/3
m
GR
1/6
m
m
Ei
Example 2.16
Birthdays are good for you.
Statistics show that the people
who have the most live the
longest.
-Reverend Larry Lorenzoni
8
2021/03/18
Example
9