notes, examples and problems presented by del ferster
TRANSCRIPT
![Page 1: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/1.jpg)
Breakout Session #1Conditional and
Combinatorial ProbabilityNotes, examples and problems presented
byDel Ferster
![Page 2: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/2.jpg)
We’ll take another look at some of the topics that were included on the exam that deal with probability.
We’ll explore the FUNDAMENTAL COUNTING PRINCIPLE.
We‘ll look at PERMUTATIONS and COMBINATIONS.
I’ve also brought along some nice lesson ideas that deal with probability, that I think you might find useful.
What’s in store for today’s session?
![Page 3: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/3.jpg)
Aren’t these kids cute!
![Page 4: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/4.jpg)
The Fundamental Counting Principle
Essential Question
How is the fundamental counting principle
applied to determine outcomes?
![Page 5: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/5.jpg)
Fundamental Counting Principle for Multi-Step Experiments
If an experiment can be described as a sequence of k steps with n1 possible outcomes on the fist step, n2 possible outcomes on the second step, then the total number of possible outcomes for the experiment is given by:
)( . . . ))(( 21 knnn
![Page 6: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/6.jpg)
Del, can you put that in other words??
Sure, it simply says that we multiply the number of ways that each component of the experiment can be achieved, and in so doing, calculate the TOTAL number of possible outcomes for the experiment.
![Page 7: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/7.jpg)
Let’s look at some Examples
Look out for things like whether outcomes can be repeated or not.
Later on, we’ll hear the words “with replacement” and “without replacement”
![Page 8: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/8.jpg)
Example #1
Del’s Deli (say that one 3 times really fast! ) features 4 kinds of breads and 6 kinds of meats. If your lunch sandwich consists of one type of bread, and one type of meat, how many different sandwiches could you build?
4 6 24 ways
![Page 9: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/9.jpg)
Example #2
Always looking to expand its offerings, Del’s Deli adds Cheese to its menu.
(yes, Packers fans, you can now get your cheese-on at Del’s Deli!)
Customers can now choose from 4 kinds of bread, 6 kinds of meat, and 5 kinds of cheese. How many sandwiches that consist of one kind of bread, one kind of meat, and one kind of cheese can be constructed?
4 6 5 120 ways
![Page 10: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/10.jpg)
Let’s look at some more challenging problem
types.Watch out for whether repetition is allowed or not allowed.
![Page 11: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/11.jpg)
Example 3
Suppose that Pennsylvania License plates have 3 letters followed by 4 digits.
How many different licenses plates are possible if digits and letters can be repeated?
26 26 26 10 10 10 10 175,760,000 ways
![Page 12: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/12.jpg)
Example 4
Consider the same PA license plate situation (3 letters followed by 4 digits)
How many different licenses plates are possible if digits and letters can NOT be repeated?
78,624,000 ways26 25 24 10 9 8 7
![Page 13: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/13.jpg)
ET Phone homeExample 5
How many different 7 digit phone numbers are possible if the 1st digit cannot be a 0 or 1?
Assume digits can be repeated.
8 10 10 10 10 10 10
8,000,000 ways
![Page 14: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/14.jpg)
ET Phone homeExample 6
How many different 7 digit phone numbers are possible if the 1st digit cannot be a 0 or 1,
AND if no repetition is allowed?
8 9 8 7 6 5 4 483,840 ways
![Page 15: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/15.jpg)
PERMUTATIONSA fancy math
word that means arrangements
![Page 16: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/16.jpg)
PERMUTATIONS
A permutation is an ordered grouping of items
Determines the number of ways you may arrange r elements from a set of n objects when order matters
![Page 17: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/17.jpg)
A formula for Permutations, and an introduction to FACTORIAL notation
!
!n r
nP
n r
If we wish to order (or arrange) r objects from an available collection of n objects, we have:
![Page 18: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/18.jpg)
A Note or 2 about factorials
! 1 2 3 2 1
0! 1
n n n n
A Factorial is the product of all the positive numbers from 1 to a number.
![Page 19: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/19.jpg)
A television news director wishes to use 3 news stories on an evening show. One story will be the lead story, one will be the second story, and the last will be a closing story. If the director has a total of 8 stories to choose from, how many possible ways can the program be set up?
Permutation Example #1
![Page 20: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/20.jpg)
A television news director wishes to use 3 news stories on an evening show. One story will be the lead story, one will be the second story, and the last will be a closing story. If the director has a total of 8 stories to choose from, how many possible ways can the program be set up?
Solution to Permutation Example #1
Since there is a lead, second, and closing story, we know that order matters. We will use permutations.
8 3
8!336
5!P
8 33
or 8 7 6 336P
![Page 21: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/21.jpg)
Permutation Example #2
A school musical director can select 2 musical plays to present next year. One will be presented in the fall, and one will be presented in the spring. If she has 9 to pick from, how many different possibilities are there?
![Page 22: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/22.jpg)
A school musical director can select 2 musical plays to present next year. One will be presented in the fall, and one will be presented in the spring. If she has 9 to pick from, how many different possibilities are there?
Solution to Permutation Example #2
Order matters, so we will use permutations.
9 2
9!72
7!P
9 22
or 9 8 72P
![Page 23: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/23.jpg)
CombinationsSelecting Items where
ORDER DOESN’T MATTER
![Page 24: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/24.jpg)
COMBINATIONS A COMBINATION is a grouping of items, WITHOUT REGARD to order
Determines the number of ways you may select r elements from a set of n objects when order doesn’t matter at all!
Sometimes we say, n choose r (we’re looking to select r items from the
n available items, without regard to order)
![Page 25: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/25.jpg)
A formula for Combinations
!
! !n r
nC
r n r
If we wish to select r objects from an available collection of n objects, we have:
![Page 26: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/26.jpg)
Combination Example #1 Dr. Ferster plans to play some serious music while he builds his next PowerPoint. He has 14 classic rock CDs to select from (including CCR, the Eagles, and Fleetwood Mac. (Sorry, no Taylor Swift or Jay Z) If Dr. F wants to select 4 CDs to play, without regard to order, how many ways can he choose his music?
![Page 27: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/27.jpg)
Dr. Ferster plans to play some serious music while he builds his next PowerPoint. He has 14 classic rock CDs to select from (including CCR, the Eagles, and Fleetwood Mac. (Sorry, no Taylor Swift or Jay Z) If Dr. F wants to select 4 CDs to play, without regard to order, how many ways can he choose his music?
Solution to Permutation Example #2
Order does not matter, so we will use combinations.
14 4
14! 14!
4! (14 4)! 4! 10!C
14 13 12 111001
4 3 2 1
![Page 28: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/28.jpg)
Combination Example #2
Dr. Ferster plans to select 5 people at random from his class of 11 students to join him at the next Packers game at Lambeau Field (Cheesehead Heaven!!)
How many ways can he select the lucky people?
![Page 29: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/29.jpg)
Dr. Ferster plans to select 5 people at random from his class of 11 students to join him at the next Packers game at Lambeau Field (Cheesehead Heaven!!)
How many ways can he select the lucky people?
Solution to Permutation Example #2
Order does not matter, so we will use combinations.
11 5
11! 11!
5! (11 5)! 5! 6!C
11 10 9 8 7462
5 4 3 2 1
![Page 30: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/30.jpg)
PROBABILITYExtending the
idea:
![Page 31: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/31.jpg)
Probability
Probability can be defined as the chance of an event occurring. It can also be used to quantify what the “odds” are that a specific event will occur.
As an aside: in VEGAS, odds are usually given against something happening.
◦For example the odds against the Packers winning the Super Bowl are now 9 to 4
![Page 32: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/32.jpg)
Probability Example #1
Mary and Frank have decided to have 3 children. Assuming that the chance of having a boy is exactly the same as having a girl, find the probability that Mary and Frank will have 2 girls and 1 boy.
![Page 33: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/33.jpg)
Using a Tree Diagram
B
G
B
G
B
G
B
G
B
G
B
G
B
G
BBB
BBG
BGB
BGG
GBB
GBG
GGB
GGG
3(2 1 )
8P Girls and Boy
![Page 34: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/34.jpg)
Kinds of probability
We’ll consider 2 types of probability
Classical Probability
Empirical Probability
![Page 35: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/35.jpg)
Classical Probability
Classical probability uses sample spaces to determine the numerical probability that an event will happen and assumes that all outcomes in the sample space are equally likely to occur.
# of desired outcomes
Total # of possible outcomes
n EP E
n S
![Page 36: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/36.jpg)
EXAMPLE A normal 6 sided die is tossed one time.
Find the probability:◦That the toss yields a 5.
◦That the toss yields an even number.
◦That the toss yields a result greater than or equal to 3.
1
63 1
6 2
4 2
6 3
![Page 37: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/37.jpg)
Empirical Probability
Empirical probability relies on actual experience to determine the likelihood of outcomes.
frequency of desired class
Sum of all frequencies
fP E
n
![Page 38: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/38.jpg)
EXAMPLE In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood.
Type FrequencyA 22B 5
AB 2O 21
Total 50
If 1 person is chosen at random from this group, find the probability:1. The person has type O
blood2. The person has type A
or type B blood.3. The person does NOT
have type AB blood.
![Page 39: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/39.jpg)
If 1 person is chosen at random from this group, find the probability:
1. The person has type O blood
2. The person has type A or type B blood.
3. The person does NOT have type AB blood.
SolutionType Frequency
A 22B 5
AB 2O 21
Total 50
21
50
27
5048
50
![Page 40: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/40.jpg)
Combining Everything
A Probability Example that makes use of combinations.
![Page 41: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/41.jpg)
Probability Example Having just won the lottery, Dr. Ferster has decided to choose 5 students at random from his Math 106 class, and fund their college education. (Don’t worry, he already has Conor and Morgan covered… ). The class is comprised of 6 boys and 4 girls.
![Page 42: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/42.jpg)
Having just won the lottery, Dr. Ferster has decided to choose 5 students at random from his Math 106 class, and fund their college education. (Don’t worry, he already has Conor and Morgan covered… ). The class is comprised of 6 boys and 4 girls.
Example (Continued)
How many ways can he choose the 5 students from his class?
10 5
10! 10!
5! (10 5)! 5! 5!C
10 9 8 7 6252
5 4 3 2 1
1
![Page 43: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/43.jpg)
Having just won the lottery, Dr. Ferster has decided to choose 5 students at random from his Math 106 class, and fund their college education. (Don’t worry, he already has Conor and Morgan covered… ). The class is comprised of 6 boys and 4 girls.
Example (Continued)How many ways can he choose the students so that he has 3 boys and 2 girls?
6 3
6! 6!
3! (6 3)! 3! 3!C
6 5 420
3 2 1
2
The BOYS
![Page 44: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/44.jpg)
Having just won the lottery, Dr. Ferster has decided to choose 5 students at random from his Math 106 class, and fund their college education. (Don’t worry, he already has Conor and Morgan covered… ). The class is comprised of 6 boys and 4 girls.
Example (Continued)How many ways can he choose the students so that he has 3 boys and 2 girls?
4 2
4! 4!
2! (4 2)! 2! 2!C
4 3
62 1
2
The GIRLS
![Page 45: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/45.jpg)
Boys Girls
6 3C
Having just won the lottery, Dr. Ferster has decided to choose 5 students at random from his Math 106 class, and fund their college education. (Don’t worry, he already has Conor and Morgan covered… ). The class is comprised of 6 boys and 4 girls.
Example (Continued)How many ways can he choose the students so that he has 3 boys and 2 girls?
4 2C
6
2
Putting it together
20 120
![Page 46: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/46.jpg)
Having just won the lottery, Dr. Ferster has decided to choose 5 students at random from his Math 106 class, and fund their college education. (Don’t worry, he already has Conor and Morgan covered… ). The class is comprised of 6 boys and 4 girls.
Example (Continued)Find the probability that Dr. F. selects 3 boys and 2 girls when he selects his 5 students
3
Putting it together
6 3 4 2
10 5
Pr(3 2 )C C
Boys and GirlsC
20 6 120Pr(3 2 ) .476
252 252Boys and Girls
![Page 47: Notes, examples and problems presented by Del Ferster](https://reader034.vdocument.in/reader034/viewer/2022042718/56649e735503460f94b72902/html5/thumbnails/47.jpg)
Wrapping Up Thanks for your attention and participation.
◦I know it’s not easy doing this after a full day with the “munchkins”.
I hope that your year is off to a good start. If I can help in any way, don’t hesitate to
shoot me an email, or give me a call.