Download Lesson 1 - Law of Large Numbers

LAW OF LARGE
NUMBERS
OBJECTIVE
•Understand the Law of
Large Numbers
•Calculate basic
probabilities.
RELEVANCE
To find the likelihood of an
event occurring by
observing the long-term
proportion in which a
certain outcome is
observed.
Simulation: Proportion of Heads
Flip the Coin
Coin Flip Simulation
10 Times
What Do You Notice?
• Short Term (Fewer Flips):
The observed proportion
of heads is different and
unpredictable for each
experiment.
50 Times
• Long Term: As the # of
5000 Times
flips increases the
proportion heads toward
50%.
Did You Notice?
• As the number of flips increased, the closer
the proportion came to the expected value
of 50%.
• This is the basic premise of probability.
Probability deals with experiments that
yield random short-term results or
outcomes yet reveal long-term
predictability.
Definition
•The long-term proportion in
which a certain outcome is
observed is the probability of
that outcome.
In Other Words
• Probability describes how likely it is that some
event will happen.
• If we look at the proportion of times an event has
occurred over a long period of time (or over a
large number of trials), we can be more certain of
the likelihood of its occurrence.
• This phenomenon is referred to as the Law of
Large Numbers.
Law of Large Numbers
Labs
Let’s see the law of large
numbers at work.
Terminology
Term
Definition
• Experiment
• Any process with uncertain
• Sample Space (S)
results that can be repeated.
• The collection of all possible
outcomes of a probability
experiment.
• Event (E)
• Any collection of outcomes
from a probability experiment.
• Unusual Event
• An event that has a low
probability of occurring.
Find the Sample Space
Event
Sample Space
Toss 1 Coin
H or T
Answer a T or F
Question
Toss 2 Coins
T or F
HH, TT, HT, TH
A probability experiment consists of
rolling a single fair die.
Identify the outcomes:
Event 1
Event 2
Event 3
Event 4
Event 5
Event 6
Roll a 1
Roll a 2
Roll a 3
Roll a 4
Roll a 5
Roll a 6
Determine the Sample Space:
𝑺 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔
Define the event E = “roll an even number”
𝟐, 𝟒, 𝟔
A probability experiment consists
having 2 children.
Identify the outcomes:
Event 1
Event 2
Event 3
Event 4
Boy, Boy
Boy, Girl
Girl, Boy
Girl, Girl
Determine the Sample Space:
𝑺 = 𝑩𝑩, 𝑩𝑮, 𝑮𝑩, 𝑮𝑮
Define the event E = “have one boy”
𝑩𝑮, 𝑮𝑩
Notation
•Use of the notation P(E)
means “the probability that
event E occurs.”
Probability Rules
Rule 1
• The probability of
Rule 2
• The sum of the
any event E, P(E),
must be between 0
and 1 inclusive.
probabilities of all
outcomes must
equal 1.
𝟎 ≤ 𝑷(𝑬) ≤ 𝟏
𝑺𝒖𝒎 𝑷(𝑬) = 𝟏
No negative probability
No probability bigger than 1.
Probability Rules
Rule 3
Example
If an event can
NEVER happen
the probability
is 0.
What is the
probability of
rolling a “9” on
a fair die?
𝑷 𝟗 =𝟎
Probability Rules
Rule 4
If an event is
CERTAIN to
happen the
probability is 1.
Example
What is the
probability of
rolling a “number
less than 7 ” on a
fair die?
𝑷 # < 𝟕 =1
Example
If a year is
selected at
random, find the
probability that
Thanksgiving Day
will be on a
Wednesday.
Thanksgiving
ALWAYS falls on
the 4th Thursday
of November.
Thus,
𝑃 𝑊𝑒𝑑 = 0
Example
What is the
probability that
New Year’s will
fall on January
1st?
ALWAYS!
𝑷 𝟏𝒔𝒕 = 𝟏
3 Probability Methods
• Empirical
• Experimental
• Classical (Theoretical)
• Assumes all outcomes in
S are equally likely.
• Subjective
• The probability of an
event is estimated by
using knowledge of the
relevant circumstances.
How are probabilities expressed?
• Probabilities are
expressed as reduced
fractions, decimals
rounded to 2 or 3
decimal places, or,
where appropriate,
percentages
Examples:
1.
1
2
2. 0.5
3. 50%
Empirical Probability
• Probabilities computed in this manner rely
on experimental evidence.
• In other words, it is evidence based on the
outcomes of a probability experiment.
𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝒐𝒇 𝑬
𝑷 𝑬 =
# 𝒐𝒇 𝑻𝒓𝒊𝒂𝒍𝒔 𝒊𝒏 𝑬𝒙𝒑𝒆𝒓𝒊𝒎𝒆𝒏𝒕
Empirical Probability Example
• If a person rolls a die
40 times and 9 of the
rolls results in a “5”,
what empirical
probability was
observed for the
event “5”?
• Answer:
9
P (5) 
 .225
40
Example: Empirical Probability
• Pass the Pigs is a
game in which pigs are
used as dice. Points
are earned based on
how the pigs land.
There are 6 possible
outcomes. 52
students rolled the pigs
3939 times. The # of
outcomes each
occurred in shown.
Find the P(Side With Dot)
Estimate the
probability that a
pig lands on a
“side with a dot.”
𝑷 𝑺𝒊𝒅𝒆 𝒘𝒊𝒕𝒉 𝒂 𝑫𝒐𝒕 = 𝟎. 𝟑𝟐𝟗
Classical Probability
• The classical method does not require
that a probability experiment actually be
performed (like the empirical).
• It relies on counting techniques.
• It requires equally likely outcomes (each
has the same probability of occurring).
Classical Probability
P(E)
S
Probability of event E
Sample Space
N(E)
Number of outcomes in E
N(S)
Number of outcomes in the sample space.
𝑵(𝑬)
𝑷 𝑬 =
𝑵(𝑺)
Classical Probability Example
What is the probability of
drawing a queen from a deck of
cards?
4
1
P(Queen ) 

52 13
Classical Probability Example
If a family has 3 children, find the
probability that all 3 children are
girls.
You are going to have to look at
the sample space before you can
answer this one.
Looking for all 3 girls
• Sample Space:
BBB
BBG
BGB
GBB
GGG
GGB
GBG
BGG
• Answer:
1
P( All 3 Girls ) 
8
Classical Probability Example
A card is drawn from an ordinary
deck. Find these probabilities:
Event
Probability
𝑷 𝑱𝒂𝒄𝒌
𝟒
𝟏
=
𝟓𝟐 𝟏𝟑
𝑷 𝟔 𝒐𝒇 𝑪𝒍𝒖𝒃𝒔
𝟏
𝟓𝟐
𝑷 𝑹𝒆𝒅 𝑸𝒖𝒆𝒆𝒏
𝟐
𝟏
=
𝟓𝟐 𝟐𝟔
Classical Probability Example
• What is the probability of rolling a
die and getting a “5”?
1
P ( E )   .17
6
Empirical vs. Classical
Empirical
Classical
• If a person rolls a die 40
• What is the probability of
times and 9 of the rolls
results in a “5”, what
empirical probability was
observed for the event
“5”?
rolling a die and getting a
“5”?
9
P (5) 
 .225
40
1
P(5)   .17
6
Subjective Probabilities
• A subjective probability of an outcome is a
probability obtained on the basis of personal
judgment. (Has high degree of personal bias)
• Ex: If a sports reporter is asked what he thinks
the chances are for the Boston Red Sox to play in
the World Series, he will make an educated
guess. He may say 20%. His probability is not
based on repeating the season. It is based on his
own personal judgment which is very subjective.
Examples of Subjective Probability
• An economist
predicts that there
is a 20% chance
of recession next
year.
• Ask New York
Yankee fans
before the start of
the baseball
season the
chances of
winning the World
Series.
Complement of an Event
• Suppose that the probability of an event E
is known and we would like to determine
the probability that E does NOT occur.
• This can be easily accomplished using the
idea of complements.
Definition
• Let S denote the sample space of a
probability experiment.
• Let E denote an event.
• The complement of E, denoted 𝑬, is all
outcomes in the sample space S that are
NOT outcomes in the event E.
Complement Rule
• If E represents any event and
𝑬 represents the complement of E, then
𝑷 𝑬 = 𝟏 − 𝑷(𝑬)
𝑬
𝑬
Example
• According to the American Veterinary Medical
Association, 31.6% of American households own a
dog.
• What is the probability that a randomly selected
household will NOT own a dog?
𝑷 𝑫𝒐 𝑵𝑶𝑻 𝒐𝒘𝒏 𝒂 𝑫𝒐𝒈 = 𝟏 − 𝑷 𝑶𝒘𝒏 𝒂 𝑫𝒐𝒈
= 𝟏 − 𝟎. 𝟑𝟏𝟔
= 𝟎. 𝟔𝟖𝟒
Example
• The probability that a person owns a
computer is 0.70.
• Find the probability that a person does NOT
own a computer.
𝑷 𝑫𝒐 𝑵𝑶𝑻 𝒐𝒘𝒏 𝒂 𝑪𝒐𝒎𝒑𝒖𝒕𝒆𝒓 = 𝟏 − 𝑷 𝑶𝒘𝒏𝒔 𝒂 𝑪𝒐𝒎𝒑𝒖𝒕𝒆𝒓
= 𝟏 − 𝟎. 𝟕𝟎
= 𝟎. 𝟑𝟎
Example
• The probability that a person DOES NOT own
a TV is
𝟏
.
𝟓
• Find the probability that a person DOES own a
computer.
𝑷 𝑫𝒐𝒆𝒔 𝑵𝑶𝑻 𝒐𝒘𝒏 𝒂 𝑻𝑽 = 𝟏 − 𝑷 𝑶𝒘𝒏𝒔 𝒂 𝑻𝑽
𝟏
=𝟏 −
=
𝟒
𝟓
𝟓
Putting It Together
2 dice are rolled. Find
a. P(sum of 3)
b. P(sum is at least 3)
c. P(sum is more than 9)
You need your array of the sums first!
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
P(sum of 3)
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
2
1
P( sum of 3) 

36 18
P(Sum is at least 3)
-Can Use the Complement!
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
P(at least 3)  1  P(less than 3)
• Prob = 1 – 1/36 = 35/36
P(sum is more than 9)
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
6
7
8
9
10 11
10 11 12
P ( more than 9) 
6
1

36
6