Download 4.5

Section 4.5
Combining Probability and Counting
Techniques
With some added content
by D.R.S., University of Cordele
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Example 4.34: Calculating Probability Using
Combinations
A group of 12 tourists is visiting London. At one
particular museum, a discounted admission is given to
groups of at least ten.
a. How many combinations of tourists can be made for
the museum visit so that the group receives the
discounted rate?
b. Suppose that a group of the tourists does get the
discount. What’s the probability that it was made up
of 11 of the tourists?
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Example 4.34: Calculating Probability Using
Combinations
A group of 12 tourists is visiting London. At one
particular museum, a discounted admission is given to
groups of at least ten.
Analysis: They will get the museum discount if they
bring how many people to the museum?
______ or ______ or ______ out of the group of 12
choose to go to the museum.
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Tourists group discount, part (a)
• How many ways to take 12 tourists 10 at a time?
_______ = _______
• How many ways to take 12 tourists 11 at a time?
_______ = _______
• How many ways to take 12 tourists 12 at a time?
_______ = _______
• Use TI-84 nCr to find these values, not the primitive
formula.
• So in all, how many ways to get “at least 10” for the
museum? ______________________
Example 4.34: Calculating Probability Using
Combinations (cont.)
b. Suppose that a group of the tourists does get the
discount. What’s the probability that it was made
up of 11 of the tourists?
Say it in conditional probability notation with words:
P ( _________________ | _________________ )
Calculate it:
Numerator = _________
Denominator = ________
or approximate decimal
value: ____________
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Example 4.35: Calculating Probability Using
Permutations
Jack is setting a password on his computer. He is told
that his password must contain at least three, but no
more than five, characters. He may use either (the 26)
letters or numbers (0–9).
a. How many different possibilities are there for his
password if each character can only be used once?
b. Suppose that Jack’s computer randomly sets his
password using all of the restrictions given above.
What is the probability that this password would
contain only the letters in his name?
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Password Permutations, continued
• It’s a lot like the tourists.
• The password can contain either _____ characters
or ____ characters or _____ characters.
• How many ways to take 36 characters ___ at a time?
• How many ways to take 36 characters ___ at a time?
• How many ways to take 36 characters ___ at a time?
• How many possible passwords in all? __________
Jack’s Password, continued.
b. Suppose that Jack’s computer randomly sets his
password using all of the restrictions given above. What
is the probability that this password would contain only
the letters in his name?
First: How many passwords are there made up only of the
characters “J”, “A”, “C”, and “K”? _____ = ______
Then: Form the probability:
Numerator: How many J, A, C, K passwords: ______
Denominator: How many possible passwords in all: _____
Rounded decimal approximation: ______________
More with JACK
What if the question were “What is the probability that
the password will contain all four letters of his name?”
• What’s “new” here, besides the passwords we
counted in the previous problem?
• How to count the number of passwords that meet
the requirements?
• Will this be more probable, less probable, or the
same probability as before?
• What is the probability now?
Example 4.36: Using Numbers of Combinations
with the Fundamental Counting Principle
Tina is packing her suitcase to go on a weekend trip.
She wants to pack 3 shirts, 2 pairs of pants, and 2 pairs
of shoes. She has 9 shirts, 5 pairs of pants, and 4 pairs
of shoes to choose from. How many ways can Tina pack
her suitcase? (We will assume that everything
matches.)
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Suitcase, continued
• Permutations or Combinations?
• ______ shirts, taken _____ at a time: __________
• ______ pants, taken _____ at a time: __________
• ______ pair of shoes, taken
_____ at a time: __________
Example 4.36: Using Numbers of Combinations
with the Fundamental Counting Principle (cont.)
The final step in this problem is different than the final
step in the last two examples, because of the word
“and.” The Fundamental Counting Principle tells us we
have to multiply, rather than add, the numbers of
combinations together to get the final result. There are
then
84  10  6  5040
different ways that Tina can pack
for the weekend.
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Systems/Quant Systems, Inc.
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Example 4.37: Using Numbers of Combinations
with the Fundamental Counting Principle
An elementary school principal is putting together a
committee of 6 teachers to head up the spring festival.
There are 8 first-grade, 9 second-grade, and 7
third-grade teachers at the school.
a. In how many ways can the committee be formed?
b. In how many ways can the committee be formed if
there must be 2 teachers chosen from each grade?
c. Suppose the committee is chosen at random and
with no restrictions. What is the probability that 2
teachers from each grade are represented?
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Example 4.37: Using Numbers of Combinations
with the Fundamental Counting Principle (cont.)
Solution
In how many ways can the committee be formed?
a. There are no stipulations as to what grades the
teachers are from and the order in which they are
chosen does not matter, so we simply need to
calculate the number of combinations by choosing
6 committee members from the 8 + 9 + 7 = 24
teachers. Calculate it here:
_____________ = ________________
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Systems/Quant Systems, Inc.
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Example 4.37: Using Numbers of Combinations
with the Fundamental Counting Principle (cont.)
b. How many ways if we get two from each grade?
How many ways
to fill the
first-grade slots?
How many ways
to fill the
second-grade slots?
How many ways
to fill the
third-grade slots?
Do you add (“or”) or multiply (“and”) ?
Example 4.37: Using Numbers of Combinations
with the Fundamental Counting Principle (cont.)
c. P(a committee does have two from each grade)
Compute the probability:
how many ways to get two from each grade
how many ways to form six overall any way
Should come up with 0.1573
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Systems/Quant Systems, Inc.
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Menu planning example
150 guests are coming.
Select 5 appetizers, 4 entrees, 3 side dishes.
Available 18 appetizers, 9 entrees, 12 side dishes.
How many different menu plans are there?
Menu planning example
• Fundamental counting principle:
• How many ways to choose appetizers,
• times how many ways to choose entrees,
• times how many ways to choose side dishes
• What about the 150 guests?
Reading list example
There are 10 novels, 8 non-fiction, and 6 self-help
books on the shelf.
Choose 5 books to take on vacation.
How many different book choices?
Reading list example, modified
There are 10 novels, 8 non-fiction, and 6 self-help
books on the shelf.
Choose 5 books to take on vacation, and at least one
book must be a self-help book.
How many different book choices?
Reading list example, modified
There are 10 novels, 8 non-fiction,18 books besides the
self-help books, and 6 self-help books on the shelf.
Choose 5 books to take on vacation, and at least one
book must be a self-help book.
• How many ways to choose 1 self-help + 4 others
• + How many ways to choose 2 self-help + 3 others
• + How many ways to choose 3 self-help + 2 others
• + How many ways to choose 4 self-help + 1 other
• + How many ways to choose 5 self-help + 0 others
Reading list example, modified
And… what if the order of the five books matters?
 Just calculate the number of ways to choose the five
books, to start with.
 Deal with “in order” in the post game show.
 Multiply the number of ways to choose the books
by 5!, since each of those ways has 5! possible
orders.
Mixture of events of different natures
How many outcomes for this statistics triathlon?
• Tossing a coin 5 times
• Then drawing 2 cards without replacement
• Then rolling a single six-sided die twice.
Mixing events of different natures
Fundamental Counting Principle, so multiply
• The coin tosses are independent events
• The card drawing is a combination
• The die rolls are independent events.
Tossing a coin 10 times
… and the number of heads is between 5 and 8
Tossing a coin 10 times
… and the number of heads is between 5 and 8
• 10 coin tosses … the sample space has ______
outcomes.
• How many ways to get 5 heads? 10 C 5
• How many ways to get 6 heads? 10 C 6
• How many ways to get 7 heads? 10 C 7
• How many ways to get 8 heads? 10 C 8