M146 - Chapter 5 Handouts - CoffeeCup Software

M146 - Chapter 5 Handouts

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Chapter 5

Objectives of chapter:

Understand probability values.

Know how to determine probability values.

Use rules of counting.

Section 5-1 Probability Rules

What is probability?

It’s the of the occurrence of some event

In plain English, it’s the of something happening

It also is the foundation for

Basic Notation:

P =

A =

P(A) =

Probability values must be between and .

Scale of probabilities:

Different ways to report probabilities:

Decimal form

Percent form

Reduced fraction


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Experiment – some “action” or process whose outcome cannot be

with certainty. Very simple examples: flipping a coin, or rolling a single die

Event – some specified or a collection of

outcomes that may or may not occur when an experiment is performed

Sample space – list of all possible for the experiment.

Probability model – list of all the possible outcomes of a probability

experiment, and each outcome’s . Note that

the sum of the probabilities of all outcomes must equal .

Unusual event – an event that has a probability of occurring.

Typically (but not always), an event with a probability is

considered to be unusual.

Example: A single coin toss Probability Model:

Sample space = Outcome Probability

Example: Toss two coins:

Sample space =

Example: Rolling a single die

Sample space =

Example: Rolling two dice


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Three methods to define the probability of an event:

1. The Empirical Method (aka Experimental)

2. The Classical Method (aka Theoretical)

3. The Subjective Method

1. Empirical Method (Experimental Probability)

Empirical evidence is evidence based on the outcomes of an

Example: From the M146 class survey:

Dominant Hand Frequency Dominant Hand Probability

Right Right

Left Left

Example: Roll two dice 100 times. Record the number of times you get

exactly one 6:

Estimate the probability of the event using the Empirical Approach:


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2. Classical Method (Theoretical Probability) The classical method calculates the probability that is

by mathematics.

It requires all of the outcomes for the experiment to be

to occur.

Examples: All of the experiments listed on page 2

Example: If you roll a single die, what is the probability of getting an even

number?

P(even) = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑖𝑡 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟

𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

Example: If you roll two dice, what is the probability of getting exactly one six?

P(one 6) = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑖𝑡 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟

𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠


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Example: If you roll two dice, what is the probability of getting exactly one six?

Experimental probability =

Theoretical probability =

Experiment Roll two dice 1000 times. Record the number of times you get

exactly one 6.

Law of Large Numbers:

States that if an experiment has a

of trials,

The experimental probability will the

theoretical probability or the probability predicted by mathematics.

Therefore, the estimate gets with more trials.

3. Subjective Method

Basically an .

Can base on past experience and current knowledge of relevant

circumstances.

Example: What is the probability that your car will not start when you try to

leave campus?

Example: What is the probability that the Seahawks will win the Superbowl

this season?


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Tree Diagrams

Useful for small problems to list the sample space

Example: Flip a coin (record H or T)

Roll a single die (record 1, 2, 3, 4, 5, or 6)

Define event as: A = T, even number

Calculate experimental probability:

Number of T/even =

Total no. of trials =

Experimental probability of A =

Calculate theoretical probability of T/even by listing the sample space:

Start:


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1. Identifying Probability Values

a. What is the probability of an event that is certain to occur?

b. What is the probability of an impossible event?

c. A sample space consists of 10 separate events that are equally likely.

What is the probability of each?

d. On a true/false test, what is the probability of answering a question

correctly if you make a random guess?

e. On a multiple-choice test with five possible answers for each questions,

what is the probability of answering a question correctly if you make a

random guess?

2. Excluding leap years, and assuming each birthday is equally likely, what is

the probability that a randomly selected person has a birthday on the 1st

day of a month?

3. What is the probability of rolling a pair of dice and obtaining a total score

of 10 or more? (Hint: look at the sample space on p. 2).

4.


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Section 5.2 The Addition Rule and Complements

Objectives:

The Addition Rule for Disjoint Events

The General Addition Rule

Complement Rule

Disjoint (or Mutually Exclusive) Events

Disjoint events have .

There is at all.


A = event that the black die is a 1

B = event that both dice are displaying the same number

C = event that the sum of the dice is more than 7

1. A & B: are they disjoint? 2. A & C: are they disjoint?


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Addition Rule for Disjoint Events

The special addition rule only applies to

events.

Calculate the probability of either event A happening event B happening

as follows:

P(A or B) =

P(A or B or C) =

Example: Christmas ornaments

(never too early to start shopping!)

Define the following events:

A = plain round ornament

B = pointy decorated ornament

If one ornament is randomly selected, find the probability that it is a plain round

ornament or a pointy decorated ornament:

P(A or B) = P(plain round or pointy decorated) =


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Example: Religion in America

Find the probability that a randomly selected American adult is Catholic or

Protestant.

P(Catholic or Protestant) =


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General Addition Rule

The general addition rule is for events that are mutually exclusive.

P(A or B) =

Example: Christmas ornaments

Define the following events:

A = round ornament

B = decorated ornament

If one ornament is randomly selected, find the probability that it is a round

ornament or a decorated ornament:

First, solve intuitively, by just looking at the picture:

P(A or B) = P(round or decorated) =

Second, solve rigorously, by applying the General Addition Rule:

P(A or B) =

Key Point: associate the word ‘or’ with of the

probabilities


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Using the General Addition Rule with Contingency Tables

A contingency table, or two-way table is used to record and analyze the

relationship between two or more variables, usually categorical variables

Example: The results of the sinking of the Titanic, which had a total of 2223

passengers

Men Women Boys Girls Total

Survived 332 318 29 27 706

Did not survive 1360 104 35 18 1517

Total 1692 422 64 45 2223

Row variable is:

Column variable is:

1. Determine the probability that a randomly selected passenger is a woman.

2. Determine the probability that a randomly selected passenger is a boy or a

girl.

3. Determine the probability that a randomly selected passenger is a man or

someone who survived the sinking.

http://www.answers.com/topic/level-of-measurement


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Complement of an Event

Every event has a complement event, which is basically the

of the event.

Event A:

Complement of A:

Notation for complement:

Example: Roll a single die

A = roll a ‘6’

Ac =

Use a Venn Diagram to show the relationship between event A and its

complement:

Complement Rule

Each of the events has an associated probability:

P(A) = probability that

P(Ac) = probability that A

Relationship between these two probabilities is: Rewrite this into the Complement Rule:


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The complement rule can be useful for simplifying calculations.

Example:

Find the probability that the religious affiliation of a randomly selected US adult is

not Jewish.

P(not Jewish) =


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Here is a standard deck of playing cards, 52 cards total, with 4 suits (spades,

hearts, clubs and diamonds). Hearts and diamonds are red, spades and clubs

are black. Each suit goes from ace, 2, 3, …, up through Jack, Queen and King,

where Jack, Queen and King are considered to be “face cards”.

1. A card is drawn at random from a deck. What is the probability that it is an

ace or a king?

2. A card is drawn at random from a deck. What is the probability that it is

NOT an ace or a king?

3. A card is drawn at random from a deck. What is the probability that it is

either a red card or an ace?


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4. A couple is planning on having three children. What is the probability that

they have three of the same gender? (Hint: try a tree diagram).

Kid 1: Kid 2: Kid 3: Start: 5.


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Section 5.3 Independence and the Multiplication Rule

1. Identify Independent Events

2. Multiplication Rule for Independent Events

3. Computing “at least” probabilities

Independence

Two events are independent if the knowledge that one event occurred does

of the other event occurring.

Two events are dependent if the occurrence of one event

the probability of another event.

Examples: Independent events

1.

2.

Example: Blocks, 4 squares, 3 triangles

1. If I randomly select one, what is the probability of selecting a square?

2. Assuming that I got a square on the first grab, what is the probability that I

reach in a second time and grab a triangle?

It !

If I : P(triangle) =

If I : P(triangle) =


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Key Point:

If sampling is done replacement, then the events are

.

If sampling is done replacement, then the events are

.

IF the events are , then can use the Multiplication

Rule for Independent Events to calculate the probability of two events happening.

Multiplication Rule for Independent Events

If A and B are independent events, then: P(A and B) =

This can be extended to multiple independent events:

P(A and B and C and …) =

Notice that this applies to .

In other words:

Event A occurs in ,

Then Event B occurs in

(and possibly more events)

Example: Roll one die, then a second die. What is the probability of getting a

1 on both?

A = get a ‘1’ on first die

B = get a ‘1’ on second die


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If a fair die (singular of dice) is rolled five times, which of the following ordered

sequences of results, if any, is MOST LIKELY to occur?

a. 3 5 1 6 2

b. 4 2 6 1 5

c. 2 2 2 2 2

d. Sequences (a) and (b) are equally likely.

e. All of the above sequences are equally likely.

Key Point: associate the word ‘and’ with

of the probabilities.

Example: Christmas lights are often designed with a series circuit. This

means that when one light burns out the entire string of lights goes

black. Suppose that the lights are designed so that the probability

a bulb will last 2 years is 0.995. The success or failure of a bulb is

independent of the success or failure of other bulbs. What is the

probability that in a string of 100 lights all 100 will last 2 years?

Computing “At-Least” Probabilities

Complement Rule: If A = “at least one” of something happens, then Ac = Example: For the Christmas lights, what is the probability that at least one

bulb will burn out in 2 years?


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2016 Presidential Election Exit Poll Results An exit poll was conducted by Edison Research during the 2016 U.S.

Presidential election, and is based on questionnaires completed by voters

leaving 350 voting places throughout the US, and also including telephone

interviews with early and absentee voters. The following table provides the

results, by race of the voters.

Voted for Clinton Voted for Trump Voted for Other/No Answer

White 37% 57% 6%

Black 89% 8% 3%

Latino 66% 28% 6%

Asian 65% 27% 8%

Other 56% 36% 8%

Sources: http://www.cnn.com/election/results/exit-polls/national/president https://www.nytimes.com/interactive/2016/11/08/us/politics/election-exit-polls.html?_r=0 Calculate the following probabilities to three significant figures.

1. If two White voters are randomly selected, what is the probability that they both

voted for Trump?

2. If three Black voters are randomly selected, what is the probability that all three

of them voted for Clinton?

3. If two Asian voters are randomly selected, what is the probability that the first

voted for Clinton and the second voted for Trump?

4. If three Latino voters are randomly selected, what is the probability that at least

one of them voted for Clinton?

http://www.cnn.com/election/results/exit-polls/national/president

https://www.nytimes.com/interactive/2016/11/08/us/politics/election-exit-polls.html?_r=0


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Section 5.4 Conditional Probability and the General Multiplication Rule

1. Compute Conditional Probabilities

2. Compute probabilities using the General Multiplication Rule

The conditional probability of an event is the probability that the event occurs,

assuming that another event has .

The conditional probability that event B occurs given that event A has occurred is

written:

Example: Face cards

Let: A = get a face card (Jack, Queen or King)

B = get a Queen

a. If one card is randomly selected, find the probability that it is a Queen.

b. If one card is randomly selected, find the probability that it is a Queen,

given that it is a face card.

Conclusion: Knowing that it is a face card the probability that

it is a Queen.


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Conditional Probability Rule:

If A and B are any two events, then:

𝑃(𝐵|𝐴) =

Example: Titanic passengers, 2223 total

Men Women Boys Girls Total

Survived 332 318 29 27 706

Did not survive 1360 104 35 18 1517

Total 1692 422 64 45 2223

Assume that one of the 2223 passengers is randomly selected.

a. Determine the probability that the passenger is a man.

b. Determine the probability that the passenger is a man, given that the

selected passenger did not survive.

Conclusion: Knowing that the passenger did not survive the

probability that it is a man.


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Defining Independent Events using Conditional Probabilities

From the Titanic example on the previous page: P(B) = P(man) = P(B | A) = P(man | did not survive) = Are the events “man” and “did not survive” independent?


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The International Shark Attack

File, maintained by the

American Elasmobranch

Society and the Florida Museum

of Natural History, is a

compilation of all known shark

attacks around the globe from

the mid 1500s to the present.

Following is a contingency table

providing a cross-classification

of worldwide reported shark

attacks during the 1900s, by

country and lethality of attack.

a. Find the probability that an attack occurred in the United States.

b. Find the probability that an attack occurred in the United States and that it

was fatal.

c. Find the probability that an attack was fatal.

d. Find the probability that an attack was fatal, given that it occurred in the

United States.

e. Find the probability that an attack occurred in the United States, given that

it was fatal.

f. Are the events “fatal” and “occurred in the United States” independent?


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IF the events are , then use the

General Multiplication Rule to calculate the probability of two events happening.

General Multiplication Rule

If A and B are any two events, then: P(A and B) =

Example: Blocks, 4 squares, 3 triangles

1. Assuming replacement after each selection, find the probability that I

randomly select two blocks (one at a time), and get a square first and a

triangle second:

P(square & triangle) =

Note: in this case, the events are independent.

2. Assuming replacement after each selection, find the

probability that I randomly select two blocks (one at a time), and get a

square first and a triangle second:

P(square & triangle) =

Note: in this case, the events are NOT independent.

Example: From a deck of cards, find the probability of selecting two cards

without replacement, and having them both be Kings.

Example: Pick two cards from a deck without replacement:

What is the probability of getting those two cards?


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Often, we assume sampling for

calculations, even though technically the sampling is done without replacement.

The reason: calculating probabilities WITH replacement is much easier, because

we don’t have to worry about the conditional probabilities changing every time we

make a selection.

Example: CBC students

Assume 7400 students total, 4144 female students

Question: If one student is selected at random, what is the probability that it is

a female student?

Question: If five different students are selected at random, what is the

probability that ALL FIVE are female students?

Calculate without replacement:

P(female & female & female & female & female) =

Now, calculate assuming with replacement:

It’s OK in this case to assume replacement, because the sample size is very

compared to the population size.


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Using the Complement Rule

Example: CBC students

If five different students are randomly selected, what is the probability of selecting

at least one student who is female?

Remember, just calculated that P(female) =

A =

Ac =


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1. Two cards are drawn from a deck without replacement. What is the

probability they are both diamonds?

2.

3. Assume that you are going to (maybe this weekend) take a quiz with 5

multiple choice questions, each with 4 possible answers. You randomly

guess. What is the probability of getting at least one question right?

(note: please do not actually use this strategy!)


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4. Use the sample data from the following table, which includes results from experiments conducted with 100 subjects, each of whom was given a polygraph test.

Polygraph Indicated Truth

Polygraph Indicated Lie

Subject actually told the truth

65 15

Subject actually told a lie 3 17

a. If 1 of the 100 subjects is randomly selected, find the probability of getting

someone who told the truth or had the polygraph test indicate that the

truth was being told.

b. If two different subjects are randomly selected, find the probability that

they both had the polygraph test indicate that a lie was being told. Do the

calculation without replacement.

c. Repeat the calculation in part b., but this time assume replacement.


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The Birthday Problem

Basic idea is to find the probability for our class that AT LEAST TWO people

have the same birthday.

Complement: people in the room have the same birthday.

Assumptions:

Assume 365 possible birthdays

Assume all birthdays are equally likely

Assume no twins (or otherwise) in the room ()

Source: Tri-City Herald, October 6, 2007


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Use the Complement Rule:

It’s easier to calculate the probability that NO TWO people in the class have the

same birthday.

A = at least two people have the same birthday (what we want)

Ac = nobody has the same birthday

P(A) = 1 – P(Ac)

P(at least 2 people have same birthday) = 1 – P(nobody has same birthday)

Calculate: P(nobody has the same birthday), or P(Ac)

1. Find the probability that TWO randomly selected people do NOT have the

same birthday:

Probability that the 1st person has a birthday =

Probability that the 2nd person’s birthday is different =

P(1st birthday AND different 2nd birthday) =

2. Find the probability that out of THREE randomly selected people, NONE

of them have the same birthday.

Probability that the 3rd person’s birthday is different from the first two =

P(1st birthday AND different 2nd birthday AND different 3rd birthday) =


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Number of people in the room: r =

3. Find the probability that out of randomly selected people,

NONE of them have the same birthday.

Probability that the person’s birthday is different from the first =

P(1st birthday AND different 2nd birthday AND different 3rd birthday … AND

different birthday) =

P(Ac) = (probability that nobody in here has same b-day)

Now take the complement of this value:

Therefore,

P(A) = P(at least two people in here have the same birthday)

= 1 – P(Ac) =


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Easier way: the probability that out of r randomly selected people, NONE of

them have the same birthday, can be represented mathematically by the formula:

Probability =

where,

r = no. of people

365Pr = no. of permutations of 365 objects taken r at a time

To calculate: 365Pr

Calculator

Commands

TI-30X

365 2nd xy _r_ 2nd nPr =

TI-30Xa

365 2nd nPr _r_ =

TI-30XIIb TI-30XIIs

365 PRB nPr (enter) _r_ =

TI-30XS 365 prb (enter) _r_ (enter)

TI-34II

365 PRB _r_ =

TI-36X

365 xy _r_ 2nd nPr =

TI-68

365 3rd nPr _r_ =

Casio fx-260 solar 365 SHIFT nPr _r_ =

Test: 365P2 = 132,860

Try to use your calculator to calculate the value we just computed.

r = Calculate r

rP

365

365 =

P(at least two people in here have the same birthday) = 1 – r

rP

365

365 =


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Section 5.5 Counting Techniques

The principal of counting means: finding how many different

an event can have.

Example: Powerball

January 2016 record Powerball drawing: $1.6 billion grand prize

If you buy a Powerball ticket, what is the probability that you will have the winning

numbers?

To calculate this, we have to know how many different ways there are to choose

the numbers.

How many outcomes when you:

Roll a single die

Roll two dice

Draw a single card

Toss two coins

For a couple having 3 children

Techniques to count:


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The Multiplication Rule (Basic Counting Rule)

Talking about making a sequence of choices from separate categories. In other

words, sequential trials, or sequential events:

Pick something from the first category,

Then pick from the second category,

Etc.

Each selection from each category can have a certain number of outcomes:

Category No. of outcomes

The Multiplication Rule says that:

To find the number of possible outcomes,

together the individual number of outcomes for

each category.

Example: Flipping two coins

Total outcomes =


Total outcomes =

Example: A couple having three kids

Total outcomes =


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Example: Assume that a license plate consists of 4 letters followed by three

digits. How many plates are possible (letters and digits may be

repeated)?

Event No. of outcomes

Pick 1st letter

Pick 2nd letter

Pick 3rd letter

Pick 4th letter

Pick 1st digit

Pick 2nd digit

Pick 3rd digit

Total no. of outcomes =

How many plates are possible if the letters and digits may NOT be repeated?

Applying Counting Rules to Probability

What’s the probability of randomly generating 4 letters and 3 digits and having it

be your plate, assuming letters and digits may be repeated?

The outcomes are equally likely, so apply the “Classical Method” to calculate

probabilities:

P =


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Factorial Notation

n! is factorial notation, and it is read ‘n factorial’.

n! =

n must be a non-negative integer

Examples:

Permutations

A permutation is: any different

of a certain number of objects.

KEY POINT: matters when you are counting up permutations!

Example:

How many different ways can the objects be arranged?


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Example:

How many different ways can two objects at a time selected from this group be

arranged?

Permutation Rule: nPr =

nPr = number of permutations

n = total number of objects

r = number of them taken at a time

Read as: “the number of permutations of n objects taken r at a time”

(example this page) 4P2 =

(example previous page) 3P3 =

Special Permutations Rule:

Just a special case of the permutation rule.

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


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Permutations with Nondistinct Items

Example: How many ways to arrange all of the objects:

(they are ALL distinct)

Example: How many ways to arrange all of the objects:

(they are NOT all distinct)

Will it be more, or less, or the same?

Formula: The number of permutations of n objects of which n1 are of one kind,

n2 are of a second kind, …, and nk are of a kth kind is given by:

Example: How many different six-digit numerals can be written using all of the

following six digits: 4, 4, 5, 5, 5, 7?


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Combinations

Combinations are different from permutations because

of the objects does not matter

Not how many different arrangements there are, just how many

different .

Example: a, b, c

Previously found that there were permutations of these objects.

How many different combinations of these objects are there?

Example: a, b, c, d

Previously found that there were permutations when selecting two

objects at a time from this group.

How many different combinations of two objects can be selected from this group?

Combinations Rule: nCr =

nCr = number of combinations

n = total number of objects

r = number of them taken at a time

Read as: “the number of combinations of n objects taken r at a time”

Example: Find the number of combinations of 25 objects taken 8 at a time.


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Selections from Two (or more) Subgroups

Example: Cracked Eggs

A carton contains 12 eggs, 3 of

which are cracked. If we randomly

select 5 of the eggs for hard

boiling, how many outcomes are

there for the following events?

a. Select any 5 of the eggs.

b. All of the cracked eggs are selected.

Note: in this case, we are choosing specific numbers from the two sub-groups,

cracked and not cracked.

1. Use the Rule to determine the number of

outcomes for the selections from each subgroup.

2. Use the Rule to multiply the individual outcomes

together.

Cracked: 3 Not cracked: 9

Select: Select: (for a total of 5)

What is the probability that all of the cracked eggs are selected?


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Example: Powerball

If you buy a Powerball ticket, how many different outcomes are there? In other

words, how many different ways to choose the numbers? Draw 5 different white

balls out of a drum that has 69 white balls and 1 red ball out of a drum that has

26 red balls in it.

White: 69 Red: 26

Select: Select:

What is the probability that you win?


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Items Selected

With Replacement

Multiplication Rule

Without Replacement

Order does

NOT matter

Combinations

Order does

matter

Permutations Multiplication

Rule

All

items

distinct?

yes

no Special Permutations

Rule (when some

items are identical to

others)

Special case: if you are

selecting or “drawing”

from 2 or more groups of

things, calculate the

combinations for each and

multiply together using

the Multiplication Rule.

Note that these two are equivalent

(when counting without

replacement), so you can use either

one (I usually do permutations).

How many total items are there? = n

How many of them are being selected? = r

Summary of Counting Methods

Two main questions: 1. Is the selection with or without replacement? 2. Does order matter?

Sequentially, from different categories

From one big “bag” of items


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Example 1: From a committee of 8 people, how many ways can we choose a

subcommittee of 2 people?

1. With or without replacement?

2. Does order matter?

Example 2: From a committee of 8 people, how many ways can we choose a

chairperson and a vice–chair?



Example 3: How many 5-card hands can be dealt from a deck of 52 cards?




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1. How many ways can a 10-question multiple choice test be answered if

there are 5 possible answers (A, B, C, D, and E) to each question? Hint:

this is kind of like the license plate problem, use the fill-in-the-blank

method!

2. Ten people gather for a meeting. If each person shakes hands with each

other person exactly once, what is the total number of handshakes?

3. In the Washington State Lotto game, players pick six different numbers

between 1 and 49. What is the probability of winning the Lotto?

4. A nurse has 6 patients to visit. How many different ways can he make his

rounds if he visits each patient only once?


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5. In a dog show, a German Shepherd is supposed to pick the correct two

objects from a set of 20 objects. In how many ways can the dog pick two

objects?

6. A slot machine consists of three wheels with 12 different objects on a

wheel (each wheel has the same 12 objects). How many different

outcomes are possible?

7. In a Jumble puzzle, you are supposed to unscramble letters to form

words. How many ways can the letters CATSITTISS be arranged? (Also,

what is the word?)

8. The Hazelwood city council consists of 5 men and 4 women. How many

different subcommittees can be formed that consist of 3 men and 2

women?

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