CHAPTER 2--PROBABILITY



Chapter 2--Probability.Doc

STATISTICS 301—APPLIED STATISTICS, Statistics for Engineers and Scientists, Walpole, Myers, Myers, and Ye, Prentice Hall

[pic]

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?

__________________________________________________________________

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%,

• and lastly sometimes as fractions, [pic].

The interpretation of a probability does not depend on how it is written, expressed, or represented.

( 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 [pic] or you have one chance in 135 million in winning so if you win you just encountered a VERY, VERY, VERY rare and unusual event!

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: probability of event A = P(A) = Pr{A}

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

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}?

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%

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?

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 15 32 0 2 50

2.00 30.00 64.00 0.00 4.00 100.00

MALE 7 13 51 1 3 75

9.33 17.33 68.00 1.33 4.00 100.00

All 8 28 83 1 5 125

6.40 22.40 66.40 0.80 4.00 100.00

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

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 }.

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 }

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.)

| | |Reason for job loss | | |

|Workplace moved/closed |Slack Work |Position abolished |Total | |Male |1,703 |1,196 |548 |3,447 | |Female |1,210 |564 |363 |2,137 | |Total |2,913 |1,760 |911 |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) =

Example #2

Example #3

Example #4

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 }

• Equally Likely Outcomes (E.L.)

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