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Organized Counting

Organized Counting. Tree Diagrams Fundamental Counting Principle Additive Counting Principle. Created by: K Wannan Edited by: K Stewart. Example 1.

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Organized Counting

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  1. Organized Counting Tree Diagrams Fundamental Counting Principle Additive Counting Principle Created by: K Wannan Edited by: K Stewart

  2. Example 1 You are making a yummy sandwich and you have a choice of white or brown bread. On your sandwich you can have only one of the following; ham, chicken or beef. To dress the sandwich you can use mustard or mayonnaise. How many different kinds of sandwiches can you make?

  3. Model: Tree Diagrams • Each stage in the sandwich making process has multiple choices. Together, different paths are made that show all of the possible outcomes (sandwiches)

  4. Total of 12 Different Sandwiches Is there a faster way to find this result than counting results in a tree diagram?

  5. Multiplicative Counting Principle(Model: Formula) • Why can we multiply the number of options in each stage to get the total number of possible outcomes (in this case, the total number of possible sandwiches)? There are 2 types of Bread There are 3 types of Meat There are 2 types of Dressing 2x3x2 = 12

  6. Example 2 At lunch in a restaurant you are given a choice of either a soup or a salad as an appetizer. There are five soups and four salads to choose from. How many different options are there for appetizers?

  7. Model: Picture By counting, we can see that there are nine choices. Is there a faster way than drawing an image and counting?

  8. Additive Counting Principle(Model: Formula) • Why can we add the number of options in each stage to get the total number of possible outcomes (in this case, the total number of possible appetizers)? There are five soups and four salads 5 + 4 = 9

  9. Example 3 • Sailing ships use signal flags to send messages. There are four different flags. • Messages are sent using at least two of these flags for each message. • Flags cannot be repeated. • How many different messages are possible?

  10. Mutually Exclusive Events • You can either fly two flags or three flags orfour flags at a time. • These actions can’t happen at the same time. ( it isn’t possible to fly only two flags and also be flying four flags simultaneously!) • These events are called MUTUALLY EXCLUSIVE events. They are separate actions that can’t happen at the same time.

  11. Mutiplicative Counting Principle and Additive Counting Principle(Model: Formula) • Sailing ships use signal flags to send messages. There are four different flags. • Messages are sent using at least two of these flags for each message. • Flags cannot be repeated. • How many different messages are possible? Determining the number of possible messages for 2, 3 and 4 flags uses the multiplicative counting principle. Why? 2 Flag Message 4X3 =12 3 Flag Message 4X3X2 = 24 4 Flag Message 4X3X2X1=24 The total number of possible messages in the three possible situations uses the additive counting principle. Why? 12 + 24 + 24 = 60 different possible messages

  12. Example • Steph has four pairs of shoes in her gym bag. How many ways can she pull out an UNMATCHED (left and right) pair of shoes? Sometimes it is easier to solve a problem indirectly. In this case we can find out how many ways it takes to find a MATCHED pair, and subtract it from the total possible matches This type of indirect reasoning is commonly called the BACK DOOR METHOD

  13. Indirect Reasoning “Back Door Method” (Model: Ordered List) Total Possible Pairs Pull #1 Pull #2 8 shoes x 7 shoes = 56 A Matched Pair Pull #1 can be any one of the shoes (left or right of one of the four pairs) and then Pull #2 has only one possibility for a match Totalling 8 ways to draw a matched pair of shoes. 56 - 8 = 48 Ways that Steph can pull an UNMATCHED pair of shoes

  14. Key Concepts • Multiplicative Counting Principle • multiply choices for each stage when it’s a multi-step situation where every choice in one stage is possible for each of the choices in the previous stage, • e.g. There are four choices for appetizer and three choices for dessert. Choose an appetizer and a dessert. • Additive Counting Principle • add number of choices in each situation together when situations are mutually exclusive • e.g. There are four choices for appetizer and three choices for dessert. Choose either an appetizer or a dessert (not both). • Indirect Reasoning “Back Door Method” • Determine the number of favourable outcomes for the opposite event and subtract from total possible number of outcomes.

  15. Homework • Page 229 # 3,5 – 17, 24

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