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Chapter 11: Probability and Statistics
11.1 Permutations and Combinations
The Fundamental Counting Principle allow us to count large numbers of possibilities quickly.
You can extend the idea to any number of choices.
Example 1: A college offers 3 different English courses, 5 different math course, 2 different art
courses, and 4 different history courses. In how many ways could a student choose 1 of each
type of course?
Frequently with counting problems you will use combinations of letters and digits.
There are ____
different digits
possible:
There are ____
letters in the
alphabet.
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Permutations
A permutation is an _______________________________ of items in a particular order.
Example 2:
How many ways can 5 Scrabble tiles be placed in a row?
Any time you have a number in the form
N x (N-1) x (N-2) x . . . x 3 x 2 x 1 you can write it as N! (read “N Factorial”)
Example 3
a) How many ways can 10 different
textbooks be arranged on a shelf?
Example 4:
Calculate the following
a)
b) How many ways can you arrange
8 different shirts on hangers in your closet?
5!
b)
15!
COUNTING THE
NUMBER OF
PERMUTATIONS
Sometimes we want
to arrange items
from a set, but we
don’t want to use
ALL the items
Example 5:
Ten students are in a race. How many ways can 10 runners finish 1st, 2nd, and 3rd? (no ties allowed)
Method 1: Fundamental Counting Principle
Method 2: nPr Formula
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Most scientific calculators have and nPr function to compute the number of permutations of n objects
taken r at a time.
TI 83
Type in “n” value
Press “MATH” button.
Arrow over to PRB
Choose option #2
Type in “r” value
Hit enter
n P r = 10 P 3
Example 6:
a)
Calculate the following
12P8
b)
20P5
Permutations with Repetition
Example 7: How many ways can you arrange the letters of the word
“PARTY” ?
Example 8: How many ways can you arrange the letters of the word
“HAPPY”?
Repetition?
If you are assigning position to
things and some of the items are
identical…
Example:
N items
x are the same
y are the same
#arrangements =
N!
x! y!
Example 9: How many ways can you arrange the letters of the word
“MISSISSIPPI” ?
Example 10:
How many ways can you make a stack of 5 blocks, 3 of which are red and 2 of which are blue?
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Combinations
A combination is a _____________________ of items in which order doesn’t
necessarily matter.
Example 11:
“How many ways can you select three tiles from a set of five?”
Example 12: Calculate the
following
a)
15C3
b)
12C9
For a TI83, the nCr command is located right under the nPr
command in the Math -> PRB menu
Example 13: For each problem, determine whether to solve using a permutation or a combination.
Then find the solution.
a) A chemistry teacher divides his class into eight groups. Each group submits one drawing of the
molecular structure of water. He will select four of the drawings to display. In how many different
ways can he select the drawings?
b) You will draw winners from a total of 25 tickets in a raffle. The first ticket wins $100. The second
ticket wins $50. The third ticket wins $10. In how many different ways can you draw the three
winning tickets?
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11.2 Probability
Probability is how _________________ an event is
to occur. The probability of an event is measured
somewhere between 0 and 1.
P(impossible) = ______
P(certain) = ______
Many times probabilities are expressed as fractions or as percentages.
Actually collecting
data:
Example 1:
Of the 60 vehicles in the teachers’ parking lot today, 15 are pickup trucks.
What is the experimental probability that a vehicle in the lot is a pickup truck?
What is the probability that a vehicle is NOT a pickup truck?
Example 2:
A softball player got a hit in 20 of her last 50 times at bat.
What is the experimental probability that she will get a hit in her next at bat?
A simulation is a model of an event.
You can use a simulation to find the experimental probability of an event.
***RANDINT HANDOUT***
From your simulation, how likely is it to pass a 10 question, 4 choice multiple choice test if you randomly
guess at the answers?
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The sample space is the set of all possible outcomes of an experiment.
When each outcome has the same chance of occurring, the outcomes are called
equally likely outcomes.
Rolling a number cube:
Sample Space:
Equally likely?
Example 3: Given a standard number cube, what is the theoretical probability of each event?
a) P(even)
b) P(7)
c) P(even or multiple of 3)
Example 4: Given a standard deck of 52 playing cards, what is the theoretical probability of each event?
Note:
13 Cards: Ace, King, Queen, Jack, 10,9,8,7,6,5,4,3,2
4 Suits: Black Suits – Clubs, Spades
Red Suits – Diamonds, Hearts
a) P(King)
b) P(Face Card)
c) P(heart or a 10)
Example 5: You open a bag of jawbreakers and find 15 red, 10 orange, 8 green, and 7 purple. What is the
theoretical probability of each event?
a) P (red or orange)
b) P(not green)
c) P(red or not orange)
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Combinatorics
When you are simultaneously choosing several things at once, you may need to use combinations to determine
the probability of an event.
Example 6: What is the theoretical probability of being dealt the following 5 card hands from a standard 52-card
deck?
a) P(exactly 2 kings)
b) P(exactly 3 kings)
c) P (all 2’s or 3’s)
c) P(5 diamonds)
Example 7: A student has a personal library of 30 zombie movies and 10 action movies. If the student
randomly grabs 6 movies to take on vacation, what are the following theoretical probabilities?
a) P(all zombie movies)
b) P(all action movies)
c) P(exactly 2 zombie movies)
d) P(exactly 4 zombie movies)
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11.3 Probability of Multiple Events
Consider the following:
I have a stack of 10 cards each with a different number from 1 to 10.
1,2,3,4,5,6,7,8,9,10
Simple Probability (1 item, 1 time)

You draw 1 card from the stack. What is the probability that the card is an even number?
Combinatorics (several items at the same time)

You choose 3 cards from the stack. What is the probability that all three cards are even?

You choose 3 cards from the stack. What is the probability of getting exactly 2 odd-numbered
cards?
Multiple Events (1 item, several trials or different items)

You roll a standard number cube and flip a coin. What is the probability of getting a even
number and heads?

You flip a coin 6 times in a row. What is the probability of getting all heads?
Dependent vs. Independent
dependent - one event affects the outcome of another
independent – events do not affect each other
Example 1:
a)
Roll a number cube. Spin a spinner.
b)
Draw a card. Without replacing it, draw a second card.
c)
Flip a coin. Flip the coin again.
d)
Select a coin from a pile. Put it back. Select a coin again.
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Probability of Independent Events
If A and B are independent events, then P(A and B) = P(A) x P(B)
Example 2: Find the probability of each of the following.
a) You are rolling a standard number cube then flipping a coin.
P( even number and head)
b) You are flipping a coin three times.
P(3 heads)
c) There are 10 diet cokes and 5 regular cokes. There are 8 bags of Doritos and 12 bags of Gardettos.
P( diet drink, Doritos)
P( regular drink, gardettos)
Mutually Exclusive Events
Two events that cannot happen at the same time.
P(A and B) = 0
Example 3: You roll a standard number cube.
P(2 and 3)
P(even and prime)
Example 4: You draw a card from a deck of 52.
P(K and Q)
P(K and heart)
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“OR”
Cannot happen at same time.
Mutually Exclusive
P(A or B) = P(A) + P(B)
CAN happen at same time.
Not Mutually Exclusive
P(A or B) = P(A) + P(B) – P(A and B)
Example 5:
Students can take one foreign language at a time.
37% of students take Spanish. 15% of students take French.
P(Spanish or French)
Example 6:
Students can take multiple language courses at the same time.
37% take Spanish. 15% take French. 5% take Spanish and French.
P(Spanish or French)
Example 7: You are working with a standard deck of 52 playing cards.
a) P(heart or a diamond)
b)
P(heart or a King)
c) P(Ace or a King)
d)
P (Ace or Black Card)
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11.4 Conditional Probability
The Monty Hall Problem
Monty Hall was the host of a classic game show
called “Let’s make a deal.”
Classic Problem:
A contestant is shown 3 possible doors.
Behind 1 door is a new car.
Behind the other two doors are goats.
The contestant gets to choose one of the doors.
After the contestant chooses a door, Monty opens up one of the OTHER doors to reveal a goat. The
contestant is then given the chance to keep the door they chose, or to switch doors. What should the
contestant choose?
Conditional Probability Exists when two events are DEPENDENT…
P( B A) = “The probability of event B, given event A”
P(1st door a car 2nd door a goat)
P(female student graduate school)
Example 1: The table shows students by gender at two- and
four-year colleges, and graduate schools, in 2005. You pick a
student at random.
a) P( female graduate school ) ?
b)
P( four  year male)
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Example 2:
Americans recycle increasing amounts through
municipal waste collection. The table shows the collection of data
for 2007.
a) What is the probability that a sample of recycled waste is paper?
b) What is the probability that a sample of recycled waste is plastic?
c) What the probability that a sample of recycled waste is glass?
There is also a formula for Conditional
Probability
Example 3: A utility company asked 50 of its customers whether they pay
their bills online or by mail. What is the probability that a customer pays the bill
online, given that the customer is male?
Example 4: Researchers asked shampoo users whether they apply
shampoo directly to their head, or indirectly with their hand. What is the
probability that a respondent applies shampoo directly to their head,
given that the respondent is female?
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NOTE:
P( B A) 
P( A and B)
 P( A and B)  P( A)  P  B A 
P( A)
Example 5: A school system compiled the following information from a survey it sent to people who were
juniors 10 years earlier.
 85% of the students graduated from high school
 Of the students who graduated from high school, 90% are happy with their jobs.
 Of the students who did not graduate from high school, 60% are happy with their jobs.
a) What is the probability that a person from the junior class 10 years ago graduated from high school and is
happy with his or her job?
b) What is the probability that a student from the junior class 10 years ago did not graduate and is happy with
his or her present job?
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11.9 Normal Distributions
Many common statistics
(human height, weight,
blood pressure) tend to
have a “normal
distribution” to their
mean (their average).
You may have heard this
called a “bell curve”
Sometimes an
extraordinary factor may cause the data to be “skewed”
Example 1: The bar graph gives the weights of
a population of brown bears. The curve shows
how the weights are normally distributed
about the mean, 11.5kg.
a) Approximately what percent of female
brown bears weigh between 100 and 129 kg.
b) Approximately what percent of female
brown bears weigh less than 120 kg?
c) The standard deviation in the weights of female brown bears is about 10 kg. Approximately what percent of
female brown bears have weights that are 1.5 standard deviations of the mean?
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Example 2: For a population of male European eels, the mean body length is 21.1 inches and the standard
deviation is 4.7 inches. Sketch a normal curve showing eel lengths at one, two, and three standard deviations
from the mean.
Example 3: The scores on an
Algebra 2 final are approximately
normally distributed with a mean of
150 and a standard deviation of 15.
a) What percentage of the students who took the test scored about 180?
b) If 250 students took the final, approximately how many scored above 135?
c) If 13.6% of the students received a B on the final, how can you describe their scores?