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Counting Outcomes and Applying Counting Principles

This introduction to counting explains the fundamental counting principle, adding principle, and compound events. It also covers mutually exclusive and inclusive events, as well as independent and dependent events. Examples and exercises are provided to practice counting outcomes and manipulating factorials.

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Counting Outcomes and Applying Counting Principles

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  1. Section 11.4:Introduction to Counting (Easy enough, right?)

  2. Let’s Count! • Count out or list all the possible outcomes from flipping a coin and then rolling a dice. A chart/table would be helpful!

  3. Let’s Count! • Count out or list all the possible ways to put 4 students into 4 different colored seats.

  4. What about 50 students in 50 different seats??? • After a while enumeration (counting each possibility) is not efficient. • That’s why we need methods to calculate the number of outcomes

  5. Fundamental Counting Principle(also called the Multiplication Principle) • Look for the word AND • If event M can occur in ‘m’ ways followed by event N that can occur in ‘n’ ways, then event M followed by event N can occur in m x n ways • These events can be independent or dependent

  6. Fundamental Counting Principle Independent Events: The first event does not affect the outcome of the next event Ex) Mary had 5 concert t-shirts and 6 pairs of pants to wear to a party. How many ways can she make an outfit from these clothes ANSWER: 5 x 6 = 30 ways Independent Events

  7. Fundamental Counting Principle Dependent Events: The first event affects the outcome of the next event and the next event needs to be adjusted (reduced) accordingly. Ex) Each player in Monopoly uses one of 6 pieces. How many ways can four people choose a board game piece? ANSWER: 6 x 5 x 4 x 3 = 360 ways Dependent Events

  8. The Adding Principle & Compound Events • Look for the word OR • A compound event is one event that can be broken down into “simple” or “sub” events. Ex) Mary may choose a snack from 3 different pieces of fruit or 2 different types of granola bars. **Mary is only picking one thing** • A compound event can be mutually exclusive or inclusive

  9. The Adding Principle & Compound Events Mutually Exclusive Simple Events: To be mutually exclusive the simple events within a compound event cannot happen at the same time. For example: Rolling a die cannot result in getting a number that is both even & odd. Ex) How many ways can you draw one card from a 52-card deck and get an 8 or a King? Answer: 4 + 4 = 8 ways • You add together the number of ways for each simple event Mutually Exclusive Events A and B are mutually exclusive.

  10. The Adding Principle & Compound Events Inclusive Simple Events To be inclusive the simple events within a compound event CAN happen at the same time. For example: A teacher selects a student to read in class. What if she selects a student who is both a boy or a sophomore. Ex) How many ways can you draw a card that is an 8 or a Heart from a 52-card deck? **There exists a card in the deck that shows both of these – the 8 of hearts. This occurrence of both events is called the OVERLAP. Inclusive Events

  11. Answer: M = Number of ways for first event to occur N = Number of ways for second event to occur M ∩ N = Number of ways they occur together, or the overlap. # of ways = M + N – (M ∩ N) 4 + 13 – 1 = 16 ways

  12. Add or Multiply??Exclusive, Inclusive, Independent, Dependent?? Example #1 Example #2 Barney is choosing a book to read at the library. On the shelf there are 3 fiction books, 5 autobiographies, and 2 history books. How many ways are there for him to choose a book? Penelope is ordering a pizza. She must choose one from each of the following: 3 types of crusts, 4 types of cheeses, and 6 types of vegetables. How many ways could she create a pizza?

  13. Add or Multiply??Exclusive, Inclusive, Independent, Dependent?? Example #3 Example #4 A scholarship committee is selecting a student from District 86 to win $1000 for college. There are 25 eligible students from Hinsdale Central and 29 eligible students from Hinsdale South. How many ways are there to award this scholarship? The senior class is voting for President and Vice-President out of 10 total candidates. How many ways are there to choose a President and then a Vice-President?

  14. Brain Break

  15. Students have to move their right foot in a clockwise circle, and then with their right pointer finger, they need to write the number 6 in the air. GO SLOW…repeat with left foot and left pointer.

  16. Let’s do some problems

  17. Phone Numbers How many 7 digit phone numbers are possible if the number cannot start with zero? 2) How many possible area codes exist if you must exclude 911 and 000? License Plates How many license plates are possible if they can only have 7 numbers? 2) Michigan license plates must have 3 numbers and 3 letters. The numbers must be next to each other and the letters must be next to each other. How many of these plates are possible?

  18. Factorials! • What does 4! mean? • Factorials can be used as a more efficient way to apply the multiplication principle. • Ex) How many ways are there to arrange 16 kids in 16 different chairs?

  19. Manipulating Factorials! Ex) Ex) Ex) 0!

  20. Challenge!! • Can you simplify:

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