m146 - chapter 5 handouts - coffeecup software
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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.
Example: Rolling two dice
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.
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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?
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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 =
Example: Rolling two dice
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?
1. With or without replacement?
2. Does order matter?
Example 3: How many 5-card hands can be dealt from a deck of 52 cards?
1. With or without replacement?
2. Does order matter?
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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?
M146 - Chapter 5 Handouts
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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?