1 chapter 3 probability 3-1 fundamentals 3-2 addition rule 3-3 multiplication rule: basics 3-4...

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Chapter 3Probability

3-1 Fundamentals

3-2 Addition Rule

3-3 Multiplication Rule: Basics

3-4 Multiplication Rule: Complements and Conditional Probability

3-5 Counting Techniques

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Objectives develop sound understanding of

probability values used in subsequent chapters

develop basic skills necessary to solve simple probability problems

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General CommentsThis chapter tends to be the most difficult one encountered in the course

Homework note:Show setup of problem even if using calculator

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Experiment – Action being performed (book uses the word procedure)

Event (E) – A particular observation within the experiment

Sample space (S) - all possible events within the experiment

3-1 Fundamentals Definitions

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Notation

P - denotes a probability

A, B, ... - denote specific events

P (A) - denotes the probability of event A occurring

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Basic Rules for Computing Probability

Let A equal an Event

P(A) =number of outcomes favorable to

event “A”total possible experimental outcomes(sample space)

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Probability Limits

The probability of an impossible event is 0.

The probability of an event that is certain to occur is 1.

0 P(A) 1

Impossibleto occur

Certainto occur

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Possible Values for ProbabilitiesCertain

Likely

50-50 Chance

Unlikely

Impossible

1

0.5

0

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Unlikely Probabilities

Examples:• Winning the lottery• Being struck by lightning• 0.0000035892• 1 / 727,235

Typically any probability less than 0.05 is considered unlikely.

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Example: Roll a die and observe a 4? Find the probability.

What is the experiment? Roll a die

What is the event A? Observe a 4

What is the sample space? 1,2,3,4,5,6

Number of outcomes favorable to A is 1.

Number of total outcomes is 6.

What is P(A)? P(A) = 1 / 6 = 0.167

Similar to #4 on hw

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Example: Toss a coin 3 times and observe exactly 2 heads?

Experiment:

toss a coin 3 times

Event (A):

observe exactly 2 heads

Sample Space:

HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

Note there are 3 outcomes favorable to the event and 8 total outcomes

P(A) = 3 / 8 = 0.375

Test problem

Similar to #6 on HW

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Law of Large Numbers

As a procedure is repeated again and again, the probability of an event tends to approach the actual probability.

This is the reason to quit while your ahead when gambling!

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P(A)

Complementary Events

The complement of event A, denoted by A, consists of all outcomes in which event A does not occur.

P(A)(read “not A”)

or P(Ac)

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P(A) = 1 – P(Ac)

Complementary Events

Property of complementary events

Example: If the probability of something occurring is 1/6 what is the probability that it won’t occur?

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Example: A study of randomly selected American Airlines flights showed that 344 arrived on time and 56 arrived late, What is the probability of a flight arriving late?

Let A = late flight Ac = on time flight

P(Ac) = 344 /(344 + 56) = 344/400 = .86

P(A) = 1 – P(Ac) = 1 - .86 = 0.14 Similar to number 11 & 12 on hw

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Rounding Off Probabilities

give the exact fraction or decimal

orround off the final result to three significant digits

Examples:

•1/3 is exact and could be left as a fraction or rounded to .333

•0.00038795 would be rounded to 0.000388

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Compound Event – Any event combining 2 or more events

Notation – P(A or B) = P (event A occurs or event B occurs or they both occur)

General Rule – add the total ways A can occur and the total way B can occur but don’t double count

3-2 Addition Rule Definitions

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• Formal Addition Rule• P(A or B) = P(A) + P(B) - P(A and B)

where P(A and B) denotes the probability that A and B both occur at the same time.

• Alternate form

P(A B) = P(A) + P(B) – P(A B)

Compound Event

U

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Definition

Events A and B are mutually exclusive if they cannot occur

simultaneously.

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Definition

Total Area = 1 Total Area = 1

P(A) P(B) P(A) P(B)

P(A and B)

Non-overlapping EventsOverlapping Events

Not Mutually Exclusive

P(A or B) = P(A) + P(B) – P(A and B)

Mutually Exclusive

P(A or B) = P(A) + P(B)

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Applying the Addition Rule

P(A or B)

Addition Rule

AreA and Bmutuallyexclusive

?

P(A or B) = P(A)+ P(B) - P(A and B)

P(A or B) = P(A) + P(B)Yes

No

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Mutually Exclusive

Example: P(A) = 2/7 and P(B) = 3/7 , P(A or B) = 5/7, are A and B mutually exclusive? Why?

Test question

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Example: You have an URN with 2 green marbles, 3 red marbles and 4 white marbles

Let A = choose red marble and B = choose white marble

1. What is the probability of choosing a red marble? P(A) = 3/9

2. What is the probability of choosing a white marble? P(B) = 4/9

3. What is the probability of choosing a red or a white? P(B or A) = 3/9 + 4/9 = 7/9

Test Questions

Why are event A and B mutually exclusive events?

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Example:

A card is drawn from a deck of cards.

1. What is the probability that the card is an ace or jack?

P(ace) + P(jack) = 4/52 + 4/52 = 8/52

2. What is the probability that the card is an ace or heart?

P(ace) + P(heart) – P(ace of hearts) = 4/52 + 13/52 – 1/52 = 16/52

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Example: Toss a coin 3 times and observe all possibilities of the number of heads

Experiment: toss a coin 3 times

Events (A): observe exactly 0 heads

(B): observe exactly 1 head

(C): observe exactly 2 heads

(D): observe exactly 3 heads

Sample Space:


Find the P(A) + P(B) + P(C) + P(D)

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Example: Toss a coin 3 times and observe all possibilities of the number of heads

Event P(event)

A = 0 heads 1/8

B = 1 head 3/8

C = 2 heads 3/8

D = 3 heads 1/8

Total 1

Probability distribution: Table of all possible events along with the probability of each event. The sum of all probabilities must sum to ONE.

Note: Events are mutually exclusive

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Example: Roll 2 dice and observe the sum

Experiment: roll 2 dice

Event (F): observe sum of 5

Sample Space: 36 elements

One 2 Two 3’s Three 4’s Four 5’s Five 6’s, One 12 Two 11’s Three 10’s Four 9’s Five 8’s

Six 7’s

Find the P(F)

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Example: Roll 2 dice and observe the sum

Event P(event)

A: Sum = 2 1/36

B: Sum = 3 2/36

And so on…….

Construct a probability distribution

Let’s Try #8 From the HW

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• Let A = select a man • Let B = select a girl

• P(A or B) = + = = 0.781

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223

Contingency Table (Titanic Mortality)

* Mutually Exclusive *

16922223

452223

17372223

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• Let A = select a woman• Let B = select someone who died.

P(A or B) = (422 + 1517 - 104) / 2223 = 1835 / 2223 = 0.825


Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223

Contingency Table(Titanic Mortality)

* NOT Mutually Exclusive * Very similar to test problem

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• How could you define a probability distribution for this data?


Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223

Contingency Table (Titanic Mortality)

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Complementary EventsP(A) & P(Ac)

P(A) and P(Ac) are mutually exclusive

P(A) + P(Ac) = 1 (this has to be true)

P(A) = 1 - P(Ac)

P(Ac) = 1 – P(A)

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Venn Diagram for the Complement of Event A

Total Area = 1

P (A)

P (A) = 1 - P (A)

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Notation: P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial)

Formal Rule

P(A and B) = P(A) • P(B) if independent (with replacement)

P(A and B) = P(A) • P(B A) if dependent (without replacement)

3-3 Multiplication Rule Definitions

will define later

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Definitions Independent Events

Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other.

Dependent EventsIf A and B are not independent, they are said to be dependent.

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TaTbTcTdTeFaFbFcFdFe

a

b

c

d

e

a

b

c

d

e

T

F

P(T) = P(c) = P(T and c) =

Tree Diagram of Test Answers

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15

110

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P (both correct) = P (T and c) 1

10 1 2

1 5

= •

MultiplicationRule

INDEPENDENT EVENTS

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Choose 2 marbles from an URN with 3 red marbles and 3 white marbles

Dependent – choose the 1st marble then choose the 2nd marble

Independent – choose the 1st marble, replace it, then choose the 2nd marble

Independence vs. Dependence

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P(B A) represents the probability of event B occurring after it is assumed that event A has already occurred (read B A as “B given A”).

= given

Notation for Conditional Probability

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Example: You have an URN with 3 red marbles and 4 white marbles


1. What is the probability of choosing a red marble? P(A)

2. What is the probability of choosing a white marble? P(B)

3. If two are chosen find the probability of choosing a white on the a second trial given a red marble was chosen 1st. P(B A)

a) Assume the 1st marble is replaced {independent}

b) Assume the 1st marble is not replaced {dependent}

Test Questions

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Let’s change things a bit….

If two are chosen and we want to find the probability of choosing a white then choosing a red marble.

So if we let: A = choose red 1st and B = choose white 2nd then we need to find P(A and B)P(A and B)

The problem here is that calculating this probability depends on what happens on the first draw. We need a rule that helps us with this.

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Formal Multiplication Rule

P(A and B) = P(A) • P(B A)

If A and B are independent events, P(B A) is really the same as P(B). Will see this in the next section.

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Applying the Multiplication Rule

P(A and B)

Multiplication Rule

AreA and B

independent?

P(A and B) = P(A) • P(B A)

P(A and B) = P(A) • P(B)Yes

No

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If two are chosen find the probability of choosing a red then choosing a white marble. In other words find P(A and B)

a) Assume the 1st marble is replaced {independent}P(A and B) = P(A) • P(B)

b) Assume the 1st marble is not replaced {dependent}P(A and B) = P(A) • P(B A)

Use as an example for #6

Test Questions

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Class Assignment – Part IYou have an URN with 3 red marbles, 7 white marbles,

and 1 green marbleLet A = choose red

B = choose whiteC = choose green

Find the following:

1. P(Ac), that is find P(not red)2. P(A or B)3. If two marbles chosen what is the probability that you

choose a white marble 2nd when a red marble was chosen first. This is, find P(B A)

4. If two marbles are chosen, find P(A and C) with replacement5. If two marbles are chosen, find P(A and C) without

replacement

Very Similar to test question #1

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Class Assignment – Part II

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Mutually Exclusive vs. Independent Events

Mutually Exclusive EventsP(A or B) = P(A) + P(B)

Independent Events P(A and B) = P(A) • P(B)

Example: if P(A) = .3, P(B)=.4, P(A or B)=.7, and P(A and B) = .12, what can you say about A and B?

Note: Test Question

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• Select two– Find P(2 women) = 422/2223 x 421/2222– Find P(2 that died) = 1517/2223 x 1516/2222

• Select one– Find P(woman and died) = 104/2223– Find P(Boy and survived) = 29 / 2223


Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223

Contingency Table(Titanic Mortality)

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3-4 Topics

Probability of “at least one”

More on conditional probability

Test for independence

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Probability of ‘At Least One’

‘At least one’ is equivalent to ‘one or more’.

The complement of getting at least one item of a particular type is that you get

no items of that type.

If P(A) = P(getting at least one), then

P(A) = 1 - P(Ac)

where P(Ac) = P(getting none)

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Probability of ‘At Least One’

Find the probability of a getting at least 1 head if you toss a coin 4 times.

P(A) = 1 - P(Ac)

where P(A) is P(no heads)

P(Ac) = (0.5)(0.5)(0.5 )(0.5) = 0.0625

P(A) = 1 - 0.0625 = 0.9375

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Conditional Probability

P(A and B) = P(A) • P(B|A)Divide both sides by P(A)

Formal definition for conditional probability

P(B|A) =P(A and B)

P(A)

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If P(B|A) = P(B)

then the occurrence of A has no effect on the probability of event B; that is, A and B are independent events.

or

If P(A and B) = P(A) • P(B)

then A and B are independent events. (with replacement)

Testing for Independence

Example: if A and B are independent, find P(A and B) if P(A) = 0.3 and P(B) = 0.6 (test question)

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3-5 Countingfundamental counting rule

two events (“mn” rule)

multiple events (nr rule)

permutationsfactorial rule

different items

not all items different

combinations

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Fundamental Counting Rule(‘mn” rule)

If one event can occur m ways and the second event can occur n ways, the events together can occur a total of m • n ways.

Example 1:Example 1: How many ways can your order a meal with 3 main course choices and 4 deserts? First list then use rule.

Main Courses

Tacos (T)

Pasta (P)

Liver & Onions (LO)

Deserts

Ice Cream (IC)

Jello (J)

Cake ©

Fruit (F)

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Fundamental Counting Rule(nr rule)

If one event that can occur n ways is repeated r times, the events together

can occur a total of nr ways.

Example 2: Example 2: How many outcomes are possible when tossing a coin 3 times? First list then use rule.

Example 3: Example 3: How many outcomes are possible when tossing a coin 20 times? Would you care to list all the outcomes this time?

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The factorial symbol ! denotes the product of decreasing positive whole numbers.

n! = n (n-1) (n-2) (n-3) • • • • • (3) (2) (1)

Special Definition: 0! = 1

Find the ! key on your calculator

Notation

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A collection of n different items can be arranged in order n! different ways.

Factorial Rule

Example:Example: How many ways can you order the letters A, B, C? List first then use rule.

Note: actually a special type of permutation, will define next

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Compare the NR and Factorial Rule

Example:Example: How many ways can you order the letters A, B, C?

a) NR _______ _________ ________ (with replacement)

b) N! _______ _________ ________ (without replacement)

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n is the number of available items (without replacement)

r is the number of items to be selected

the number of permutations (or sequences) is

Permutations Rule(when items are all different)

Order matters

Pn r = (n - r)!n!

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Example: Eight men enter a race. In how many ways can the first 4 positions be determined?

Permutations Rule(when items are all different)

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Permutations Rule( when some items are identical )

If there are n items with n1 alike, n2 alike, . .

. . nk alike, the number of permutations is

n1! . n2! .. . . . . . . nk! n!

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Permutations Rule( when some items are identical )

1. How many ways can you arrange the word statistics? Or Mississippi?

Examples:

2. How many ways can you arrange 3 green marbles and 4 red marbles?

Test Question

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Permutations Rule

Factorial rule is special case

n! = n nP

Can you show this is true?

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n different items

r items to be selected

different orders of the same items are not counted (order doesn’t matter)

the number of combinations is

(n - r )! r!n!

nCr =

Combinations Rule

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TI-83 CalculatorCalculate n! , nPr, nCr

1. Enter the value for n

2. Press Math

3. Cursor over to Prb

4. Choose 2: nPr or 3: nCr or 4: n! as required

5a. Press Enter for the n! case

5b. Enter the value for “r” for the nPr and nCr

cases

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Pick five numbers from 1 to 47 and a MEGA number from 1 to 27

Pick five numbers from 1 to 56 and a MEGA number from 1 to 46

Note: game has 2 separate sets of numbers

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Example: Find the probability of winning the Pennsylvania Super 6 lotto. Select 6 numbers from 69.

What’s the probably of getting 5 of 6? 4 of 6?, etc. (see lottery handout)

What’s the probability if you have to get all 6 numbers in a specified order?

Combinations Rule

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Recall previous example?

Experiment: Toss a coin 3 times

Event (A): Observe 2 heads

Sample Space:


Note there are 3 outcomes favorable to the event and 8 total outcomes

P(A) = 3 / 8 = 0.375

Test problem

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Let’s take a different approach



There are 23 possible outcomes

and 3C2 ways to get 2 heads

P(A) = 3C2 / 23 = 3 / 8 = 0.375

Test problem

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Example:



There are 26 possible outcomes

and 6C4 ways to get 4 heads

P(A) = 6C4 / 26 = 15 / 64 = 0.234

Test problem

1 chapter 3 probability 3-1 fundamentals 3-2 addition rule 3-3 multiplication rule: basics 3-4...

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