Notes, examples and problems presented by Del 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.

How is the fundamental counting principle applied to determine outcomes?

If an experiment can be described as a sequence of k steps with n 1 possible outcomes on the fist step, n 2 possible outcomes on the second step, then the total number of possible outcomes for the experiment is given by:

 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.

 Look out for things like whether outcomes can be repeated or not.  Later on, we’ll hear the words “with replacement” and “without replacement”

 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?

 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?

Watch out for whether repetition is allowed or not allowed.

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?

 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?

 How many different 7 digit phone numbers are possible if the 1 st digit cannot be a 0 or 1?  Assume digits can be repeated.

 How many different 7 digit phone numbers are possible if the 1 st digit cannot be a 0 or 1,  AND if no repetition is allowed?

A fancy math word that means arrangements

 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

If we wish to order (or arrange) r objects from an available collection of n objects, we have:

A Factorial 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? Since there is a lead, second, and closing story, we know that order matters. We will use permutations.

 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.

Selecting Items where ORDER DOESN’T MATTER

 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)

If we wish to select r objects from an available collection of n objects, we have:

 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.

 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.

Extending 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

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.

B G B G B G B G B G B G B G BBB BBG BGB BGG GBB GBG GGB GGG

 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.

 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.

Empirical probability Empirical probability relies on actual experience to determine the likelihood of outcomes.

 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. TypeFrequency A22 B5 AB2 O21 Total 50 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.

 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. TypeFrequency A22 B5 AB2 O21 Total 50

A Probability Example that makes use of combinations.

 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.

How many ways can he choose the 5 students from his class? 1

 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. How many ways can he choose the students so that he has 3 boys and 2 girls? 2

 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. How many ways can he choose the students so that he has 3 boys and 2 girls? 2

BoysGirls  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. How many ways can he choose the students so that he has 3 boys and 2 girls? 2

 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. Find the probability that Dr. F. selects 3 boys and 2 girls when he selects his 5 students 3

 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 , or give me a call.