1. 13 Lesson 1 Let Me Count the Ways Fundamental Counting Principle, Permutations & Combinations CP Probability and Statistics FA 2014 S-ID.1S-CP.3S-CP.5

2. Objectives  Describe the Law of Large Numbers  Model Probability  Apply the Fundamental Counting Principle  Compute permutations  Compute combinations  Distinguish permutations vs combinations

3. The OR Rule: To find the probability of two independent events "E" and "F", simply add the probabilities of the two independent events: P(E or F) = P(E) + P(F)

4. The AND Rule: To find the probability of two independent events "E" and "F", simply multiply the probabilities of the two independent events: P(E and F) = P(E) * P(F)

5. Fundamental Counting Principle Lets start with a simple example. A student is to roll a die and flip a coin. How many possible outcomes will there be? 1H 2H 3H 4H 5H 6H 1T 2T 3T 4T 5T 6T 12 outcomes 6*2 = 12 outcomes

6. Fundamental Counting Principle Fundamental Counting Principle can be used determine the number of possible outcomes when there are two or more characteristics. Fundamental Counting Principle states that if an event has m possible outcomes and another independent event has n possible outcomes, then there are m * n possible outcomes for the two events together.

7. Law of Large Numbers As the # of independent trials increases, The long-run relative frequency of repeated events gets closer and closer to the value we will call the probability.

8. Sample Space Examples of Sample Space: The sample space from rolling the number cube is (1, 2, 3, 4, 5, 6) When you toss a coin, the sample space is (heads, tails) How would you define sample space?

9. It’s Just Rolls of the Dice A die is rolled 3 times. What is the probability of getting a "1" on the first roll, a "2" on the second roll, and a "3" on the third roll? "Solution" P(1) = 1/6P(2) = 1/6P(3) = 1/6 P(1,2,3) = (1/6)(1/6)(1/6) P(1,2,3) = 1/216 =.0046 =.46% Not a very good chance of happening is it??

10. Can I Guess the Right Answer? A test consists of 5 "true - false“ questions. If the person answering the questions has no idea what the correct answer is to any of the questions, what is the probability of getting them all correct? "Solution" P(true) = ½P(false) = 1/2 The probability of getting each question right or wrong is the same P(5 correct) = (1/2)(1/2)(1/2)(1/2)(1/2) P(all 5 correct) = 1/32 =.03125 = 3.125%

11. It’s In The Cards A player picks a card from a standard 52 card deck, records the card, places it back in the deck and then chooses a second card. Find the probability of choosing an ace and then a face card. "Solution" P(ace) = 4/52 = 1/13 P(face card*) = 12/52 = 3/13 P(ace,face card) = (1/13)(3/13) P(ace, face card) = 3/169 =.01775 = 1.775%

12. Fundamental Counting Principle For a college interview, Robert has to choose what to wear from the following: 4 slacks, 3 shirts, 2 shoes and 5 ties. How many possible outfits does he have to choose from? 4*3*2*5 = 120 outfits

13. Permutations A Permutation is an arrangement of items in a particular order. Notice, ORDER MATTERS! To find the number of Permutations of n items, we can use the Fundamental Counting Principle or factorial notation.

14. Permutations The number of ways to arrange the letters ABC: ____ ____ ____ Number of choices for first blank ? 3 ____ ____ 3 2 ___ Number of choices for second blank? Number of choices for third blank? 3 2 1 3*2*1 = 6 3! = 3*2*1 = 6 ABC ACB BAC BCA CAB CBA

15. Permutations To find the number of Permutations of n items chosen r at a time, you can use the formula

16. Permutations A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated? Practice: Answer Now

17. Permutations A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated? Practice:

18. Permutations From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled? Practice: Answer Now

19. Permutations From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled? Practice:

20. Combinations A Combination is an arrangement of items in which order does not matter. ORDER DOES NOT MATTER! Since the order does not matter in combinations, there are fewer combinations than permutations. The combinations are a "subset" of the permutations.

21. Combinations To find the number of Combinations of n items chosen r at a time, you can use the formula

22. Combinations To find the number of Combinations of n items chosen r at a time, you can use the formula

23. Combinations To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible? Practice: Answer Now

24. Combinations To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible? Practice:

25. Combinations A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions? Practice: Answer Now

26. Combinations A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions? Practice:

27. Combinations A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards? Practice: Answer Now

28. Combinations A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards? Practice: Center: Forwards:Guards: Thus, the number of ways to select the starting line up is 2*10*6 = 120.

29. Terms Random Phenomenon Law of Large Numbers Probability Independence (informally) Trial Outcome Event Sample Space Fundamental Counting Principle Permutations Combinations Equally Likely Condition

30. Rock, Paper Scissors – HW 4 people are playing Rock, Paper, Scissors – find the following 1 - How many different combinations are there (based on each of the 4 players having 3 options)? 2 - What are the odds of 1 person winning a round? 3 - What are the odds of a 2 person tie? 4 - What are the odds of a 3 person tie? 5 - What are the odds of no winner (at least 1 rock, 1 paper and 1 scissor are thrown)?