Notes, examples and problems presented by Del Ferster.

Breakout Session #1Conditional and

Combinatorial ProbabilityNotes, examples and problems presented

byDel Ferster

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?

Aren’t these kids cute!

The Fundamental Counting Principle

Essential Question

How is the fundamental counting principle

applied to determine outcomes?

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

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.

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”

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

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

Let’s look at some more challenging problem

types.Watch out for whether repetition is allowed or not allowed.

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

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

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

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

PERMUTATIONSA fancy math

word that means arrangements

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

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:

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.

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

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

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?

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?


Order matters, so we will use permutations.

9 2

9!72

7!P

9 22

or 9 8 72P

CombinationsSelecting Items where

ORDER DOESN’T MATTER

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)

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:

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?

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?


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

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?

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?


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

PROBABILITYExtending the

idea:

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

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.

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

Kinds of probability

We’ll consider 2 types of probability

Classical Probability

Empirical Probability

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

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

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

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.

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

Combining Everything

A Probability Example that makes use of combinations.

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.

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


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



4 2

4! 4!

2! (4 2)! 2! 2!C

4 3

62 1

2

The GIRLS

Boys Girls

6 3C



4 2C

6

2

Putting it together

20 120


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

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.

Notes, examples and problems presented by Del Ferster.

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