Counting – Day 1 Section 4.7

Why Do We Need Counting Methods? When finding a basic probability, what are the two things we need to know? Number of different simple events Number of ways event A can occur P(A) = Sometimes however, it is not practical to construct a list of the outcomes. So finding the total often requires the methods of this section.

Identity Theft A criminal (Mr. Llorens) is found using your social security number and claims that all of the digits were randomly generated. What is the probability of getting your social security number when randomly generating digits? Is the criminal’s (Mr. Llorens) claim that your number was randomly generated likely to be true?

What We Need … The Fundamental Counting Rule For a sequence of two (or more) events in which the first can occur m ways and the second can occur n ways, the events together can occur m∙n ways. Apply it! What is the probability of Mr. Llorens randomly generating your social security number?

Teapot History Facts Consider the following question given on a history test: Arrange the following event in chronological order. a) Boston Tea Party b) The invention of the teapot c) The Teapot Dome Scandal d) Ms. Yedinak buying her first teapot e) The Civil War Assuming you make a random guess, what is your probability of getting the correct answer?

Additional Information The solution to the previous problem represents a solution that can be generalized by using the following notation and factorial rule. Notation The factorial symbol (!) denotes the product of decreasing positive whole numbers. For example, 4! = 4∙3∙2∙1 = 24. By special definition, 0! =1. Factorial Rule A collection of n different items can be arranged in order n! different ways. (This factorial rule reflects that the first item may be selected n different ways, the second item may be selected n-1 ways, and so on.

Summer Vacay During your upcoming summer vacation, you are planning to visit these six national landmarks: Ms. Yedinak’s birthplace, childhood home, high school, college, favorite playground, and go-to grocery store. You would like to plan the most efficient route so you can spend as much time at these landmarks as possible. How many different route are there? TRAVESTY NEWS FLASH What if you only have time to visit 4 of these locations? How many possible routes are there now?

Permutations Rule When Items Are All Different When Some Items Are Identical To Others 1. There are n different items. 2. We select r of the n items (without replacement). 3. We consider rearrangements of the same items to be different. 1. There are n different items, and some are the same. 2. We select all of the n items (without replacement). 3. We consider rearrangements of the same items to be different. IMPORTANT: ORDER MATTERS

Examples² Horse RacingGender Selection The 132 nd running of the Kentucky Derby had a field of 20 horses. If a better randomly selects two of those horses to come in 1 st and 2 nd, what is the probability of winning? 14 couples tried to have baby girls by using the Microsort gender selection method. How many ways can 11 girls and 3 boys be arranged in sequence?

What Now? How many ways can 3 class representatives be chosen from a group of 12 students? VS.

Combinations When Order Doesn’t Matter 1. There are n different items available. 2. We select r of the n items (without replacement). 3. We consider rearrangements of the same items to be the same. (The combination ABC is the same as CBA).

Solve it! How many ways can 3 class representatives be chosen from a group of 12 students? VS.

Homework p #16, 18, 19, 25, 26