Download CHAPTER 2--PROBABILITY

Chapter 2--Probability.Doc
STATISTICS 301—APPLIED STATISTICS, Statistics for Engineers and Scientists, Walpole, Myers, Myers, and Ye, Prentice Hall
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Goal:
In this section we will tackle the concept of PROBABILITY.
Define: ____________________________________________________________
Interpretation of Probability: small prob  _____________________________
medium prob  _____________________________
large prob  ________________________________
Assigning Probability—there are 3 Methods
1. The SUBJECTIVE Method
Let’s find out how you feel about the question of intelligent life in the universe.
a. Do you believe in life elsewhere? Yes ______ No ______ Not Sure ______
b. What do you think is the probability of intelligent life in the universe? ______
2. The ASSUMPTION Method
Answer the two questions below if the
instructor rolls a six-sided die.
a. The probability of getting a “2” on the
top face is _________
b. What assumption did you make in
answering 2a ?
________________________________
________________________________
3. The FREQUENTIST Method
How would you answer the #2 if your assumption in #2 were violated?
__________________________________________________________________
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1. Probability
Defn
Probability is a number that presents the likelihood of some event happening.

Probabilities are numbers between 0 and 1. Sometimes they will be denoted by:


decimal numbers, such as 0.04
as percentages, 4%,
4
.
100
The interpretation of a probability does not depend on how it is written,
expressed, or represented.


and lastly sometimes as fractions,
Probabilities are interpreted as to whether an event is likely to occur or not based
on how close the probability is to zero or 1.
 a probability of 0 means the event will NEVER happen
 a probability of 1 means the event will always happen.
 a probability in between zero and 1 indicate that the event is less or more likely
to happen depending upon how close to zero and one it is.
Some Examples

Suppose you flip a fair coin. The probability of getting a Head is 0.50 or 50%. If we
flipped the coin once, it would not be unusual or unexpected to get a head or to not
get a head.

Suppose we roll a fair die—this means that the die has six sides and each side has the
same probability of coming up. The probability of any one of the six faces of the die
coming up is 1 out of 6 or about 0.1667 or 17%. Hence, when we roll this die and a “5”
comes up, we should not be too surprised, since 5 is not that unusual an event.

Suppose you purchase one Mega Million Lottery ticket in the hopes of winning the $73
million top prize. The chances of winning the top prize in Mega Millions purchasing
only one ticket is 0.000000007399. Note that this is pretty small number when
written as a decimal. Even written as a percentage, 0.0000007399% it is still small
and pretty hard to interpret. However, if we write it as a fraction, we obtain
1
or you have one chance in 135 million in winning so if you win you just
135,145,920
encountered a VERY, VERY, VERY rare and unusual event!
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Notation and Terminology
Sample Space
Defn:
A SAMPLE SPACE is the collection of all outcomes/elements (O1, O2, … ) from
an “experiment”
Notation: S = {
Examples
}
Toss a coin: S =
Roll a die: S =
Select a card from a poker deck: S =
Select a student from STA
: S=
Sample Spaces are not unique! Consider the following example: Toss three coins
(penny, nickel, and dime). Define two different sample spaces for this experiment.
1.
S=
2. S =
Event
Defn:
An EVENT is a collection or subset of outcomes/elements from S
Notation:
use capital Latin letters, E, A, C, etc
Examples
Toss a coin: H = Heads =
Roll a die: E = Even Number =
Select a card from a poker deck: F = Face Card =
Select a student from STA
: F = Female Student =
Probability
Notation:
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probability of event A = P(A) = Pr{A}
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More on Events (Rules concerning Combinations of Events)
Intersection
Defn:
The INTERSECTION is the set of outcomes/elements common to all events
Notation:
A  B and means
BOTH A AND B happen
Examples
Union
Defn:
The UNION is the set of outcomes/elements in any or all of the events
Notation:
A  B and means A happens OR B happens
Mutually Exclusive (M.E.)
Defn:
Events are M.E. iff A  B = empty or null set = φ, ie the set of NO
ELEMENTS
Example:
Complement
Defn:
The COMPLEMENT of an event, A, is the set of outcomes/elements in S not
in A
Notation:
A’ or Ac or Ā
Facts
1. Outcomes/Elements in S are M.E.
2. Union of the outcomes/elements from an experiment = S
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2. Probability Rules
BASIC PROBABILITY FACTS
1. Pr{A} ≥ 0 for any event
2. If A and B are M.E. events then Pr{ A  B } = Pr{ A } + Pr{ B }
3. Pr{ S } = 1
1’. With 3), 1) is usually rewritten as 0 ≤ Pr{ A } ≤ 1
Example #1
Select one M & M from a bag and note its color
S=
What are the
Probabilities of each
outcome in S?
Are the probabilities
legitimate?
Pr{ Green or Red M & M}?
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Example #2
Select a single card from a “well-shuffled,” (Why?) standard poker deck
S=
What are the Probabilities of each outcome in S? Why?
Are the probabilities legitimate?
Equally Likely Outcomes (E.L.)
Defn:
Outcomes in S are E.L. iff Pr{ any outcome } = 1 / number of outcomes in S
Use:
If an experiment yields E.L. outcomes,
then Pr{ A } = # outcomes in A / # outcomes in S
so finding probabilities amounts to counting.
Example
Since the card experiment yields E.L. outcomes,
Pr{ face card } = # face cards / # cards in deck = 12 / 52
Pr{ heart } = # hearts / # cards in deck = 13 / 52 = 1 /4 = 0.25 = 25%
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RULES/THEOREMS/RESULTS CONCERNING PROBABILITY (aka Shortcuts)
The following are a series of results that allow you to find probabilities for one event,
using probabilities of other events.
1. Pr{ A } = 1 – Pr{ A’ } = 1 – Pr{ Ac }= 1 - Pr{ Ā }
2. Addition Rule: Pr{ A  B } = Pr{ A } + Pr{ B } - Pr{ A  B }
Example:
Select a card at random from a well shuffled poker deck. Let A = face card
and B = red suited card. What is the probability we get a face card or a red
suited card?
3. Independent Events
Defn: A and B are INDEPENDENT EVENTS iff Pr { A  B } = Pr { A } * Pr { B }
Example 1:
Select a card at random from a well shuffled poker deck. Let A = face card
and B = red suited card. Are A and B independent?
Example 2: Toss a dime and a nickel. Are the outcomes on the dime and nickel
independent? Why?
Example 3: Toss a penny twice. Are the outcomes on the different tosses
independent? Why?
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RULES/THEOREMS/RESULTS CONCERNING PROBABILITY (Continued)
4. Conditional Probability
Example:
Using Class data, here is the table of Gender & Hair Color using MTB
Tabulated statistics: GENDER, HAIR
Rows: GENDER
Columns: HAIR
BLACK
BLONDE
BROWN
OTHER
RED
All
FEMALE
1
2.00
15
30.00
32
64.00
0
0.00
2
4.00
50
100.00
MALE
7
9.33
13
17.33
51
68.00
1
1.33
3
4.00
75
100.00
All
8
6.40
28
22.40
83
66.40
1
0.80
5
4.00
125
100.00
Cell Contents:
Count
% of Row
Select a student at random, let BL = get a “blonde haired student” and F = female
Pr { BL } =
Pr { F } =
Pr { BL  F } =
Lastly, what is the probability of Blonde from the Females?
Defn: CONDITIONAL PROBABILITY is the probability of one event, A, GIVEN
that another event, B, has already occurred and is denoted by Pr { A | B } =
probability of A given B
Notes
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1.
Pr { A } is actually a conditional probability, namely, Pr { A | S }, but is
never written as such!
2.
The probability of a Blonde from the Females was a conditional probability,
namely, Pr { BL | F }.
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4. Conditional Probability (Continued)
Finding Conditional Probability
Pr { A | B } = Pr { A  B } / Pr { B }, AS LONG AS Pr { B } > 0
Example: Using the Gender & Hair Color data
5. Multiplication Rule:
Pr{ A  B } = Pr{ A } * Pr{ B | A } OR
Pr{ B } * Pr{ A | B }
6. Some Miscellaneous Facts Concerning Probability
a. A and A’, its complement, are M.E.
b. A  A’ = S
c. Let B be any other event, then ( B
 A ) and ( B  A’ ) are M.E.
d. ( B  A )  ( B  A’ ) = B
e. Since Pr{ A } = 1 – Pr{ A’ } also means Pr{ A | S } = 1 – Pr{ A’ | S }, then
Pr{ A | B } = 1 – Pr{ A’ | B }
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Example #1
The table below shows the number of jobs lost (in thousands) in the United States over a
three year period. (There were 5,584,000 jobs lost.)
Workplace
moved/closed
1,703
1,210
2,913
Male
Female
Total
Reason for job loss
Slack Work
Position abolished
Total
1,196
564
1,760
548
363
911
3,447
2,137
5,584
Suppose that a person is randomly selected from the group of all persons who lost their jobs
over the three year period.
(a) What is the probability that the person is male?
Experiment: Randomly select a person
Outcome: Person selected (Outcomes are equally likely.)
Define A to be the event the person is male
P(A) =
(b) Given that the person selected lost the job due to the fact that the position was
abolished, what is the probability that the person is male?
Define B to be the event the position was abolished
P(A|B) =
(c) Given that the person selected is male, what is the probability that he lost the job
because the position was abolished?
P(B|A) =
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Example #2
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Example #3
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Example #4
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SUMMARY OF PROBABILITY
1. INTERSECTION, UNION, COMPLEMENTS, AND MUTUALLY EXCLUSIVE EVENTS.
2. BASIC PROBABILITY RULES

0 ≤ Pr{ A } ≤ 1

If A and B are M.E. events then Pr{ A  B } = Pr{ A } + Pr{ B

Pr{ S } = 1
3. RULES/THEOREMS/RESULTS CONCERNING PROBABILITY

Pr{ A } = 1 – Pr{ A’ } = 1 – Pr{ Ac }= 1 - Pr{ Ā }

Addition Rule: Pr{ A  B } = Pr{ A } + Pr{ B } - Pr{ A  B }

A and B are independent events iff Pr { A  B } = Pr { A } * Pr { B }

Pr { A | B } = Pr { A  B } / Pr { B }, AS LONG AS Pr { B } > 0

Multiplication Rule:
Pr{ A  B } = Pr{ A } * Pr{ B | A } OR Pr{ B } * Pr{ A | B }
4. Miscellaneous Facts Concerning Probability

A and A’, its complement, are M.E.

A  A’ = S

Let B be any other event, then ( B  A ) and ( B  A’ ) are M.E.

( B  A )  ( B  A’ ) = B

Since Pr{ A } = 1 – Pr{ A’ } also means Pr{ A | S } = 1 – Pr{ A’ | S }, then
Pr{ A | B } = 1 – Pr{ A’ | B }

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Equally Likely Outcomes (E.L.)
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