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3-1 Fundamentals

3-2 Addition Rule

3-3 Multiplication Rule: Basics

3-4 Multiplication Rule: Complements and Conditional Probability

3-5 Counting Techniques

Chapter 3Probabilitydevelop sound understanding of probability values used in subsequent chapters

develop basic skills necessary to solve simple probability problems

ObjectivesThis chapter tends to be the most difficult one encountered in the course

Homework note:

Show setup of problem even if using calculator

General CommentsExperiment – 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

P - denotes a probability

A, B, ...- denote specific events

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

Notation

Let A equal an Event

Basic Rules for

Computing Probability

number of outcomes favorable to event “A”

P(A) =

total possible experimental outcomes(sample space)

The probability of an impossible event is 0.

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

0 P(A) 1

Probability LimitsCertain

to occur

Impossible

to occur

Examples:

Winning the lottery

Being struck by lightning

0.0000035892

1 / 727,235

Typically any probability less than 0.05 is considered unlikely.

Unlikely Probabilities

What is the experiment? Roll a die

What is the event A? Observe a4

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

Example:Roll a die and observe a 4? Find the probability.Similar to #4 on hw

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

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

Similar to #6 on HW

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!

Law of Large Numbers

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

P(A)

(read “not A”)

P(A)Complementary Events

or P(Ac)

Property of complementary eventsP(A) = 1 – P(Ac)

Complementary Events

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

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

give the exact fraction or decimal

or

Rounding Off Probabilities- round off the final result to three significantdigits

- Examples:
- 1/3 is exact and could be left as a fraction or rounded to .333
- 0.00038795 would be rounded to 0.000388

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

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 Bboth occur at the same time.

Alternate form

P(A B) = P(A) + P(B) – P(A B)

Compound Event

Definition

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)

Total Area = 1

Total Area = 1

P(A) P(B)

P(A) P(B)

P(A and B)

Overlapping Events

Non-overlapping Events

Applying the Addition Rule

P(A or B)

Addition Rule

Are

A and B

mutually

exclusive

?

Yes

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

No

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

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

Test question

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

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

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

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

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

Why are event A and B mutually exclusive events?

A card is drawn from a deck of cards.

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

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

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

Example: 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:

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

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

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

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

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)

Example:Roll 2 dice and observe the sumLet A = select a man

Let B = select a girl

P(A or B) = + == 0.781

Contingency Table

(Titanic Mortality)

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223

* Mutually Exclusive *

45

2223

1692

2223

1737

2223

Let A = select a woman

Let B = select someone who died.

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

= 1835 / 2223 = 0.825

Contingency Table

(Titanic Mortality)

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223

* NOT Mutually Exclusive *

Very similar to test problem

How could you define a probability distribution for this data?

Contingency Table

(Titanic Mortality)

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223

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)

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

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 Events

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

DefinitionsChoose 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

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

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

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

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)

Assume the 1st marble is replaced {independent}

Assume the 1st marble is not replaced {dependent}

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

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)

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.

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

Formal Multiplication RuleApplying the Multiplication Rule

P(A and B)

Multiplication Rule

Are

A and B

independent

?

Yes

P(A and B) = P(A) • P(B)

No

P(A and B) = P(A) • P(B A)

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

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

Assume the 1st marble is replaced {independent}

P(A and B) = P(A) • P(B)

Assume the 1st marble is not replaced {dependent}

P(A and B) = P(A) • P(B A)

Test Questions

Example:You have an URN with 3 red marbles and 4 white marblesUse as an example for #6

Let A = choose red

B = choose white

C = choose green

Find the following:

P(Ac), that is find P(not red)

P(A or B)

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)

If two marbles are chosen, find P(A and C) with replacement

If two marbles are chosen, find P(A and C) without replacement

Class Assignment – Part IYou have an URN with 3 red marbles, 7 white marbles, and 1 green marbleVery Similar to test question #1

Mutually Exclusive Events

P(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?

Mutually Exclusive vs. Independent EventsNote: Test Question

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

Contingency Table

(Titanic Mortality)

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223

‘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.

Probability of ‘At Least One’If P(A) = P(getting at least one), then

P(A) = 1 - P(Ac)

where P(Ac) = P(getting none)

Find the probability of a getting at least 1 head if you toss a coin 4 times. Probability of ‘At Least One’

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

P(A and B) = P(A) • P(B|A)

Divide both sides by P(A)

Formal definition for conditional probability

Conditional ProbabilityP(A and B)

P(B|A) =

P(A)

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

fundamental counting rule

two events (“mn” rule)

multiple events (nr rule)

permutations

factorial rule

different items

not all items different

combinations

3-5 Counting(‘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: How many ways can your order a meal with 3 main course choices and 4 deserts? First list then use rule.

(nr rule)

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

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

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

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

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

Factorial Rule

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

Compare the NR and Factorial Rule

- Example: How many ways can you order the letters A, B, C?
- NR _______ _________ ________ (with replacement)
- N! _______ _________ ________ (without replacement)

nis the number of available items (without replacement)

ris the number of items to be selected

the number of permutations (or sequences) is

Permutations Rule

(when items are all different)

P

n!

n r

=

(n - r)!

- Ordermatters

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

Permutations Rule

(when items are all different)

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

Permutations Rule

( when some items are identical )

- If there are n items with n1 alike, n2 alike, . . . .nk alike, the number of permutations is

( when some items are identical )

Examples:

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

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

Test Question

n different items

r items to be selected

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

Combinations Rule

- the number of combinations is

n!

nCr=

(n - r )! r!

TI-83 Calculator

Calculate n! , nPr, nCr

- Enter the value for n
- Press Math
- Cursor over to Prb
- Choose 2: nPr or3: nCr or4: n! as required

5a. Press Enter for the n! case

5b. Enter the value for “r” for the nPr andnCr cases

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

- 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?

Experiment: Toss a coin 3 times

Event (A): Observe 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

Recall previous example?Test problem

Experiment: Toss a coin 3 times

Event (A): Observe 2 heads

There are 23possible outcomes

and 3C2ways to get 2 heads

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

Let’s take a different approachTest problem

Experiment: Toss a coin 6 times

Event (A): Observe 4 heads

There are 26possible outcomes

and 6C4ways to get 4 heads

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

Example:Test problem

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