1. (13 – 1) The Counting Principle and Permutations Learning targets: To use the fundamental counting principle to count the number of ways an event can happen To use permutations to count the number of ways an event can happen The Fundamental Counting Principle (also known as the multiplication rule for counting) If a task can be performed in n 1 ways, and for each of these a second task can be performed in n 2 ways, and for each of the latter a third task can be performed in n 3 ways,..., and for each of the latter a k th task can be performed in n k ways, then the entire sequence of k tasks can be performed in n 1 n 2 n 3... n k ways.

2. I do: (ex) A sandwich stand has four kinds of meat, ham, turkey, roast beef, and chicken. Also they have three kinds of bread, white, wheat, and rye. How many choices do you have? Ham Turkey Roast beef Chicken White Wheet Rye White Wheet Rye White Wheet Rye White Wheet Rye 3 choices with ham 3 choices with turkey 3 choices with roast beef 3 choices with chicken Total 12 choices

3. You do: (ex) You have 5 T-shirts and 3 pair of pants. How many choices of outfit do you have? We do: (ex) You can choose 3 different flowers out of 7 with the same price. How many choices you have? The first choice out of 7: 7 choices The second choice out of 6: 6 choices The third choice out of 5: 5 choices Total choices: (7)(6)(5)=210 choices

4. (13 – 1) continued Using Permutations Definition: A permutation an order of n objects. (Order is the matter) It is the same as the Fundamental Counting Principle. Notation: The number of permutations of r objects taken from a group of n distinct objects is given by: Note: A symbol ! is called “factorial”. n! = n (n – 1)(n – 2)(n – 3)…(1) (ex) 5! = 5 (4) (3) (2) (1) = 120 Remember: 0! = 1

5. I do (ex) What is the total number of possible 4-letter arrangements of the letters m, a, t, h, if each letter is used only once in each arrangement? 1 st letter4 choices 2 nd letter3 choices 3 rd letter2 choices 4 th letter1 choice Total4! = 24 arrangements

6. We do (ex) Number plate The number plate of a car consists of 3 letter and 4 numbers. How many ways can they arrange without repetition? 1 st letter: 26 choices 2 nd letter: 25 choices 3 rd letter: 24 choices AND 1 st number 10 choices 2 nd number 9 choices 3 rd number 8 choices 4 th number 7 choices Total arrangements:

8. You do: (ex) Joleen is on a shopping spree. She buys six tops, three shorts and 4 pairs of sandals. How many different outfits consisting of a top, shorts and sandals can she create from her new purchases? A pizza shop runs a special where you can buy a large pizza with one cheese, one vegetable, and meat for $9.00. You have a choice of 7 cheeses, 11 vegetables, and t meats. Additionally, you have a choice of 3 crusts and 2 sauces. How many different variations of the pizza special are possible?

9. Permutation with repetition (distinguishable permutation): The number of distinguishable permutations of n objects where one object is repeated q 1 times, another is repeated q 2 times, and so on is: I do: (ex) Find the number of distinguishable permutations of the letter in Cincinnati? Cincinnati has 10 letters. (n) i is repeated 3 times (q 1 ) n is repeated 3 times (q 2 ) c is repeated 2 times (q 3 ) =

10. We do: (ex) Find the number of distinguishable permutations of the letter in Mississippi? Mississippi has 11 letters. (n) i is repeated 4 times (q 1 ) s is repeated 4 times (q 2 ) p is repeated 2 times (q 3 )

11. You do: (ex) Find the number of distinguishable permutations of the letter in Calculus? Find the number of distinguishable permutations of the letters in the word. a)PUPPYb) LETTERc) MISSOURI d) CONNECTICUT

12. Combinations Definition: A combination is a selection of r objects from a group of n objects where the order is not important. Notation: and

13. I do: (ex) There are 12 boys and 14 girls in Mrs. Brown’s math class. Find the number of ways Mrs. Brown can select a team of 3 students from the class to work on a group project. The team is to consist of 1 girl and 2 boys. It is a combination question because order, or position, is not important. To select 2 boys from 12 is: n = 12 and r = 2 AND To select 1 girl from 14 is: n = 14 and r = 1 Total ways:

14. We do: (ex) You can select at most 3 items out of 10 items on your pizza. How many different types of pizza do you have as your choices? At most means: 0, 1, 2, OR 3 items The choice of 0 items: The choice of 1 item: The choice of 2 items: The choice of 3 items Total choices: You add if the situation is OR.

15. You do: (ex) In how many ways can 7 cards be dealt from a deck of 52 cards if you want 4 red and 3 black cards. (Hint: how many red cards and black cards are there? What is n then?) In how many ways can you obtain all 7 red or all 7 black?

16. (12 – 2) continued Using Binomial Theorem Complete the next three row of Pascal's Triangle.

17. Where I do: (ex) Expand

18. We do: (ex) Expand You do: (ex) Expand

19. (13 – 3) Probability Definition: The probability of an event is a number between 0 and 1 that shows the chance will happen. Theoretical probability (probability): When an event A will happen out of all outcomes happen equally, likely. Theoretical probability is:

20. I do: (ex) What is the probability to get an odd number when you roll a die? A die has 6 faces (1~6) An odd number are 1, 3, or 5

21. We do (ex) You shuffle the numbers 1~6 and place them. What is the probability to get 6 as the 1 st number? Total number to fill 6 places: 6! 1 st place2 nd place3 rd 4 th 5 th 6 th 6 We do: (ex) Teresa and Julia are among 10 students who have applied for a trip to Washington, D.C. Two students from the group will be selected at random for the trip. What is the probability that Teresa and Julia will be the 2 students selected?

22. You do (ex) There are 20 cards (from 1 through 20). What is the probability to get a perfect square is chosen? (1) What is the probability to get a face card out of 52-card deck if you randomly pick one card. (2) A math teacher is randomly distributing 15 rulers with centimeter labels and 10 rulers without centimeter labels. What is the probability that the first ruler she hands out will have centimeter labels and the second ruler will not have labels?

23. (3) One bag contains 2 green marbles and 4 white marbles, and a second bag contains 3 green marbles and 1 white marble. If Trent randomly draws one marble from each bag, what is the probability that they are both green?

24. Complement of probability Definition: The complement of event is that not in the event Notation: A’ is called “complement of event A” Probability of complement of A is: P(A’) = 1 – P(A)

25. I do: (ex) Two dice are tossed. What is the probability that the sum is not 8? P (sum is not 8) = 1 – P (sum is 8) Sum is 8: (2 + 6), (3 + 5), (4 + 4), (5 + 3), (6 + 2) total 5 ways Each die has 6 faces and there are two, so total # of event when two dice are tossed: 6 2 Then and

26. We do: (ex) On a certain day the chance of rain is 80% in San Francisco and 30%in Sydney. Assume that the chance of rain in the two cities is independent. What is the probability that it will not rain in either city? A 7%B 14%C 24%D 50% (0.2)(0.7) = 0.14

27. We do: (ex) The probabilities that Jamie will try out for various sports and team positions are shown in the chart below. Jamie will definitely try out for either basketball or baseball, but not both. The probability that Jamie will try out for baseball and try out for catcher is 42%. What is the probability that Jamie will try out for basketball? A 40% B 60% C 80% D 90%

28. Two six-sided dice are rolled. Find the probability of the given event. a)The sum is greater than or equal to 5 b)The sum is less than or equal to 10. c)The sum is greater than 2.

29. Probability of Independent & Dependent Events Definition: An independent event is an event that is not affected by other event. If A and B are independent events, then the probability that both A and B occur is: P(A and B) = P(A)  P(B). Probability of Independent events

30. I do (ex) You toss a die twice. What a probability to get 1 on the first toss and 3 on the second toss. We do (ex) A basket has 5 red, 4 yellow, and 3 green. You are picking 3 balls with replacement. If you pick Red, Yellow, and Green in order, you will be the winner. What is the probability to be a winner? 1 st pick2 nd pick3 rd pick Therefore,

31. You do (ex) You randomly select two cards from a 52-card deck. What is the probability that the first card is not a face and second card is a face card with replacement. One bag contains 2 green marbles and 4 white marbles, and a second bag contains 3 green marbles and 1 white marble. If you randomly draw one marble from each bag, what is the probability that they are both green?

32. Probability of Dependent events: Definition: A dependent event is an event in which one occurrence affects the other occurrence. The probability that B will occur under that A has occurred is called “Conditional probability” and noted by: P(B  A) read as “probability of B given A” Formula: If A and B are dependent events, then the probability that both A and B occur is: P(A and B) = P(A)  P(B  A)

33. I do (ex) (from CST sample question 2010) A box contains 7 large red marbles, 5 large yellow marbles, 3 small red marbles, and 5 small yellow marbles. If a marble is drawn at random, what is the probability that it is yellow, given that it is one of the large marbles? P( yellow| large) =

34. We do (ex) You and two friends are at a restaurant for lunch. There are 8 dishes with the same price. What is the probability that each of you orders a different dish? Let three dishes are A, B, and C 1 st person’s2 nd person’s 3 rd person’s

35. You do (ex) A math teacher is randomly distributing 15 plastic rulers and 10 wooden rulers. What is the probability that the first ruler she hands out will be a wooden one, and the second ruler will be a plastic one? 1 st pick2 nd pick

36. (13 – 3 continued) Probability Involving AND and OR I do (ex) Find the probability to draw a diamond or a face card from a deck of cards. 13 diamond cards : P(D) 12 face cards: P(F) 3 diamond and face cards : P(D & F) P(D or F) = P(D) + P(F) – P(D & F) =

37. We do: One card is drawn from a deck of cards. What is the probability that the card is a black or an ace? P(B) P(A) P(B & A) P(B or A)

38. You do (ex) One card is drawn from a deck of cards. What is the probability that the card is either a face or heart? P(F) P(H) P(F & H) P(F or H)

39. 1.Mean ( ): 2.Median: 3.Standard deviation: ia a measure of how each value in a data set varies from the mean. And the notation is  (sigma) orWhere x i is each data as Statistics: Definition: Statistics are numerical values used to summarize and compare sets of data. Measures of central tendency are:

40. I do (ex) Find the mean and the standard deviation for the values: 48.0, 53.2, 52.3, 46.6, 49.9 Mean Standard deviation:

41. We do (ex) Keith found the mean and standard deviation of the set of the numbers given below. If he adds 5 to each number, what is the standard deviation? 3, 6, 2, 1, 7, 5 For the data given: After 5 is added:

42. Find the mean and the standard deviation for each set of values. (a)5, 6, 7, 3, 4, 5, 6, 7, 8(b) 13, 15, 17, 18, 12, 21, 10