Formula Sheet About Exam Study Tips Hub: Math Teacher Resources Index Old Exams Graphing Calculator Guidelines SED Links Math A Topic Headings

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1 Regents Prep Math A Formula Sheet About Exam Study Tips Hub: Math Teacher Resources Index Old Exams Graphing Calculator Guidelines SED Links Math A Topic Headings Work is underway to create sites for the new Algebra, Geometry and Algebra2/Trig. Temporary draft site for new Algebra: (1 of 7)1/7/ :27:06 PM

2 Regents Prep Math A Error in Measurement 6. Uncertainty (Probability) (5-10% of Regents Exam) Theoretical versus Empirical Probability Problems Involving Probability Single & Compound Events Problems Involving AND & OR Computing Probabilities Mutually Exclusive & Independent Events Counting Principle Sample Space Tree Diagrams Permutations & Combinations Factorial Notation Permutations: npn & npr Combinations: ncn & ncr 7. Patterns & Functions (15-25% of Regents Exam) Represent & Analyze Functions Solve Linear Equations with Integral, Fraction, or Decimal Coefficients * Solve Linear Inequalities * Solve Factorable Quadratic Equations * Graphs of Linear Equations: Slope & Intercept * Graph Inequalitites * Solve Systems of Linear Equations * (6 of 7)1/7/ :27:06 PM

3 Regents Math A: Single & Compound Events Oswego City School District Regents Exam Prep Mathematics A Single & Compound Events L = Lesson, P = Practice, T = Teacher Resource Type Resource Title Author L Single & Compound Events Donna Roberts P Working with Single and Compound Events Donna Roberts T Hitting a Target Donna Roberts Copyright Oswego City School District Regents Exam Prep Center RegentsPrep and StudyZone are FREE educational resources. 12:28:22 PM

4 Regents Math A: Problems Involving AND & OR Oswego City School District Regents Exam Prep Mathematics A Problems Involving AND & OR L = Lesson, P = Practice, T = Teacher Resource Type Resource Title Author L Probability Involving AND & OR Donna Roberts P Working with Probabilities Involving AND & OR Donna Roberts T Demonstration of Working with AND & OR Donna Roberts T Simulation of Blackjack Mike McGrath Copyright Oswego City School District Regents Exam Prep Center RegentsPrep and StudyZone are FREE educational resources. 12:28:22 PM

5 Regents Math A: Mutually Exclusive & Independent Events Oswego City School District Regents Exam Prep Mathematics A Mutually Exclusive & Independent Events L = Lesson, P = Practice, T = Teacher Resource Type Resource Title Author L Complement of an Event Donna Roberts L Mutually Exclusive Events Donna Roberts L Independent Events Donna Roberts P Working with Events Donna Roberts T Hit That Calendar! Donna Roberts Copyright Oswego City School District Regents Exam Prep Center RegentsPrep and StudyZone are FREE educational resources. 12:28:22 PM

6 Regents Math A: Counting Principle Oswego City School District Regents Exam Prep Mathematics A Counting Principle L = Lesson, P = Practice, T = Teacher Resource Type Resource Title Author L The Counting Principle Donna Roberts P Working with the Counting Principle Donna Roberts T Applied Problems for the Counting Principle Donna Roberts Copyright Oswego City School District Regents Exam Prep Center RegentsPrep and StudyZone are FREE educational resources. 12:28:23 PM

7 Regents Math A: Sample Space Oswego City School District Regents Exam Prep Mathematics A Sample Space L = Lesson, P = Practice, T = Teacher Resource Type Resource Title Author L Sample Spaces Donna Roberts P Working with Sample Spaces Donna Roberts T Sample Spaces - Pair Share with a Twist! Donna Roberts Copyright Oswego City School District Regents Exam Prep Center RegentsPrep and StudyZone are FREE educational resources. 12:28:23 PM

8 Regents Math A: Tree Diagrams Oswego City School District Regents Exam Prep Mathematics A Tree Diagrams L = Lesson, P = Practice, T = Teacher Resource Type Resource Title Author L Tree Diagrams Donna Roberts P Working with Tree Diagrams Donna Roberts T Tree Diagrams - Are All Arrangements the Same? Donna Roberts Copyright Oswego City School District Regents Exam Prep Center RegentsPrep and StudyZone are FREE educational resources. 12:28:24 PM

9 Regents Math A: Factorial Notation Oswego City School District Regents Exam Prep Mathematics A Factorial Notation L = Lesson, P = Practice, T = Teacher Resource Type Resource Title Author L Factorials: Lesson Lisa Schultzkie P Factorials: Practice Lisa Schultzkie T Rolling Out Some Factorials Donna Roberts Copyright Oswego City School District Regents Exam Prep Center RegentsPrep and StudyZone are FREE educational resources. 12:28:24 PM

10 The Counting Principle Math A The Counting Principle When dealing with the occurrence of more than one event, it is important to be able to quickly determine how many possible outcomes exist. For example, if ice cream sundaes come in 5 flavors with 4 possible toppings, how many different sundaes can be made with one flavor of ice cream and one topping? Rather than list the entire sample space with all possible combinations of ice cream and toppings, we may simply multiply 5 4 = 20 possible sundaes. This simple multiplication process is known as the Counting Principle. (1 of 3)1/7/ :38:59 PM

11 The Counting Principle The Counting Principle works for two or more activities. A coin is tossed five times. How many arrangements of heads and tails are possible? By the Counting Principle, the sample space (all possible arrangements) will be = 32 arrangements of heads and tails. Remember: The Counting Principle is easy! Simply MULTIPLY the number of ways each activity can occur. (2 of 3)1/7/ :38:59 PM

12 The Counting Principle Roberts Copyright Oswego City School District Regents Exam Prep Center (3 of 3)1/7/ :38:59 PM

13 Practice with the Counting Principle Math A Working with the Counting Principle Answer the following questions using the Counting Principle. 1. A movie theater sells 3 sizes of popcorn (small, medium, and large) with 3 choices of toppings (no butter, butter, extra butter). How many possible ways can a bag of popcorn be purchased? Choose one: (1 of 6)1/7/ :41:33 PM

14 Practice with the Counting Principle 2. Information about girls' ice skates: Colors: white, beige, pink, yellow, blue Sizes: 4, 5, 6, 7, 8 Extras: tassels, striped laces, bells Assuming that all skates are sold with ONE extra, how many possible arrangements exist? Choose one: (2 of 6)1/7/ :41:33 PM

15 Practice with the Counting Principle 3. Your state issues license plates consisting of letters and numbers. There are 26 letters and the letters may be repeated. There are 10 digits and the digits may be repeated. How many possible license plates can be issued with two letters followed by three numbers? Choose one: (3 of 6)1/7/ :41:33 PM

16 Practice with the Counting Principle 4. The ice cream shop offers 31 flavors. You order a doublescoop cone. In how many different ways can the clerk put the ice cream on the cone if you wanted two different flavors? Choose one: (4 of 6)1/7/ :41:33 PM

17 Practice with the Counting Principle 5. Heather has finally narrowed her clothing choices for the big party down to 3 skirts, 2 tops and 4 pair of shoes. How many different outfits could she form from these choices? Choose one: Roberts (5 of 6)1/7/ :41:33 PM

18 Practice with the Counting Principle Math A Working with the Counting Principle Answer the following questions using the Counting Principle. 1. A movie theater sells 3 sizes of popcorn (small, medium, and large) with 3 choices of toppings (no butter, butter, extra butter). How many possible ways can a bag of popcorn be purchased? Choose one: (1 of 6)1/7/ :41:33 PM

19 Practice with the Counting Principle 2. Information about girls' ice skates: Colors: white, beige, pink, yellow, blue Sizes: 4, 5, 6, 7, 8 Extras: tassels, striped laces, bells Assuming that all skates are sold with ONE extra, how many possible arrangements exist? Choose one: (2 of 6)1/7/ :41:33 PM

20 Practice with the Counting Principle 3. Your state issues license plates consisting of letters and numbers. There are 26 letters and the letters may be repeated. There are 10 digits and the digits may be repeated. How many possible license plates can be issued with two letters followed by three numbers? Choose one: (3 of 6)1/7/ :41:33 PM

21 Practice with the Counting Principle 4. The ice cream shop offers 31 flavors. You order a doublescoop cone. In how many different ways can the clerk put the ice cream on the cone if you wanted two different flavors? Choose one: (4 of 6)1/7/ :41:33 PM

22 Practice with the Counting Principle 5. Heather has finally narrowed her clothing choices for the big party down to 3 skirts, 2 tops and 4 pair of shoes. How many different outfits could she form from these choices? Choose one: Roberts (5 of 6)1/7/ :41:33 PM

23 Sample Space Sample Space Math A A sample space is a set of all possible outcomes for an activity or experiment. Activity Rolling a die Sample Space {1, 2, 3, 4, 5, 6} Tossing a coin { Heads, Tails} (1 of 4)1/7/ :37:55 PM

24 Sample Space Drawing a card from a standard deck {52 cards} Drawing one marble from the bottle {8 marbles} Rolling a pair of dice {(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)} (2 of 4)1/7/ :37:55 PM

25 Sample Space Choosing an outfit from a green blouse, a red blouse, a black skirt, a pair of sneakers, and a pair of sandals {green blouse-skirt-sneakers, green blouse-skirt-sandals, red blouse-skirt-sneakers, red blouse-skirt-sandals} As you can see, sample spaces can become very large. When determining a sample space, be sure to consider ALL possibilities. Oftentimes this can be a difficult task. To make this process easier, we may wish to use the Counting Principle or a Tree Diagram. Remember: A sample space is the set of ALL possible outcomes. (3 of 4)1/7/ :37:55 PM

26 Sample Space Roberts Copyright Oswego City School District Regents Exam Prep Center (4 of 4)1/7/ :37:55 PM

27 Practice with Sample Spaces Math A Working with Sample Spaces Answer the following questions dealing with sample spaces. 1. How many elements are in the sample space of rolling one die? List the sample space. Choose: (1 of 4)1/7/ :41:28 PM

28 Practice with Sample Spaces 2. How many elements are in the sample space of tossing 3 pennies? List the sample space. Choose: Alarm clocks are sold in blue or pink with either digital or standard displays. How many different arrangements of alarm clocks are possible? List the sample space. Choose: (2 of 4)1/7/ :41:28 PM

29 Practice with Sample Spaces 4. You have gone to the ASPCA to adopt a puppy. You would like a beagle or cocker spaniel, that is brown or black, and has either a white tail or a brown tail. How many possible puppies fit your criteria? List the sample space. Choose: (3 of 4)1/7/ :41:28 PM

30 Practice with Sample Spaces 5. There are 2 entry doors and 3 staircases in your school. How many ways are there to enter the building and go to the second floor? List the sample space. Choose: Roberts Copyright Oswego City School District Regents Exam Prep Center (4 of 4)1/7/ :41:28 PM

31 Tree Diagrams Tree Diagrams Math A When attempting to determine a sample space (the possible outcomes from an experiment), it is often helpful to draw a diagram which illustrates how to arrive at the answer. One such diagram is a tree diagram. In addition to helping determine the number of outcomes in a sample space, the tree diagram can be used to determine the probability of individual outcomes within the sample space. The probability of any outcome in the sample space is the product (multiply) of all possibilities along the path that represents that outcome on the tree diagram. Example: Show the sample space for tossing one penny and rolling one die. (H = heads, T = tails) (1 of 4)1/7/ :39:16 PM

32 Tree Diagrams By following the different paths in the tree diagram, we can arrive at the sample space. Sample space: { H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6 } The probability of each of these outcomes is 1/2 1/6 = 1/12 [The Counting Principle would verify that this answer yields the correct number of outcomes: 2 6 = 12 outcomes.] Example 2: A family has three children. How many outcomes are in the sample space that indicates the sex of the children? Assume that the probability of male (M) and the probability of female (F) are each 1/2. (2 of 4)1/7/ :39:16 PM

33 Tree Diagrams Sample space: { MMM MMF MFM MFF FMM FMF FFM FFF } There are 8 outcomes in the sample space. The probability of each outcome is 1/2 1/2 1/2 = 1/8. [Again, the Counting Principle would verify that this result yields the correct number of outcomes: = 8 outcomes.] (3 of 4)1/7/ :39:16 PM

34 Tree Diagrams Roberts Copyright Oswego City School District Regents Exam Prep Center (4 of 4)1/7/ :39:16 PM

35 Practice with Tree Diagrams Math A Working with Tree Diagrams Answer the following questions pertaining to tree diagrams. 1. You are at a carnival. One of the carnival games asks you to pick a door and then pick a curtain behind the door. There are 3 doors and 4 curtains behind each door. How many choices are possible for the player? Choose: (1 of 4)1/7/ :41:47 PM

36 Practice with Tree Diagrams 2. The 4 aces are removed from a deck of cards. A coin is tossed and one of the aces is chosen. What is the probability of getting heads on the coin and the ace of hearts? Draw a tree diagram to illustrate the sample space. See Tree Diagram Choose: 1/2 1/4 1/8 1/16 3. There are 3 trails leading to Camp A from your starting position. There are 3 trails from Camp A to Camp B. How many different routes are there from the starting position to Camp B? Draw a tree diagram to illustrate your answer. See Tree Diagram Choose: (2 of 4)1/7/ :41:47 PM

37 Practice with Tree Diagrams 4. A spinner has 4 equally likely regions numbered 1 to 4. The arrow is spun twice. What is the probability that the spinner will land on a 1 on the first spin and on a red region on the second spin? Draw a tree diagram to represent your answer. Choose: 1/2 1/4 1/6 1/8 See Tree Diagram 5. There are two identical bottles. One bottle contains 2 green balls and 1 red ball. The other contains 2 red balls. A bottle is selected at random and a single ball is drawn. What is the probability that the ball is red? Choose: 1/2 1/3 1/6 2/3 Answer (3 of 4)1/7/ :41:47 PM

38 Practice with Tree Diagrams Roberts Copyright Oswego City School District Regents Exam Prep Center (4 of 4)1/7/ :41:47 PM

39 Single and Compound Events Math A Single and Compound Events A single event involves the use of ONE item such as: one card being drawn one coin being tossed one die being rolled one person being chosen Example: From a normal deck of 52 cards, what is the probability of choosing the queen of clubs? The deck contains only one queen of clubs, so the probability will be 1/52. (1 of 3)1/7/ :38:18 PM

40 Single and Compound Events A compound event involves the use of two or more items such as: two cards being drawn three coins being tossed two dice being rolled four people being chosen Examples: 1. Amy has 5 tank tops, 3 pairs of jeans, and 2 pairs of sneakers. How many different outfits consisting of one tank top, one pair of jeans, and one pair of sneakers are possible? Using Counting Principle: 5 x 3 x 2 = 30 outfits 2. How many different 4 letter words can be formed from the letters in the word MATH? Using Permutations: 4 P 4 = = If there are 8 marbles in a jar ( 5 are red and 3 are blue), and 4 of them are selected at random, what is the probability that 1 is blue and 3 are red? Using Combinations: (2 of 3)1/7/ :38:18 PM

41 Single and Compound Events Roberts Copyright Oswego City School District Regents Exam Prep Center (3 of 3)1/7/ :38:18 PM

42 Practice with Single and Compound Events Math A Working with Single and Compound Events Answer the following questions dealing with probability. 1. Which of the following illustrates working with compound events? Choose: rolling a die tossing 2 coins drawing one card choosing one person (1 of 5)1/7/ :41:29 PM

43 Practice with Single and Compound Events 2. A standard deck of 52 cards is shuffled. What is the probability of choosing the 5 of diamonds? Choose: 1/5 1/13 5/52 1/52 (2 of 5)1/7/ :41:29 PM

44 Practice with Single and Compound Events 3. A spinner contains eight regions, numbered 1 through 8. The arrow has an equally likely chance of landing on any of the eight regions. If the arrow lands on the line, it is spun again. What is the probability that the arrow lands on an odd number? Choose: 1/2 1/4 1/8 5/8 4. Burger Queen offers 4 types of burgers, 5 types of beverages, and 3 types of desserts. If a meal consists of 1 burger, one beverage and one dessert, how many possible meals can be chosen? Choose: (3 of 5)1/7/ :41:29 PM

45 Practice with Single and Compound Events 5. There are 12 horses in a horseshow competition. The top three winning horses receive money. How many possible money winning orders are there for a competition with 12 horses? Choose: (4 of 5)1/7/ :41:29 PM

46 Practice with Single and Compound Events Roberts Copyright Oswego City School District Regents Exam Prep Center (5 of 5)1/7/ :41:29 PM

47 Practice with Sum of Interior Angles Practice Page Math A Practice with Sum of Interior Angles Directions: Choose the correct answer from the choices listed. 1. The sum of the interior angles of a hexagon equals: 360º 540º 720º (1 of 4)1/7/ :41:31 PM

48 Practice with Sum of Interior Angles 2. How many degrees are there in the sum of the interior angles of a nine sided polygon? 1080º 1260º 1620º 3. If the sum of the interior angles of a polygon equals 900 0, how many sides does the polygon have? (2 of 4)1/7/ :41:31 PM

49 Practice with Sum of Interior Angles How many sides does a polygon have if the sum of its interior angles is 2160 º? What is a polygon called if the sum of its interior angles equals ? Octagon Decagon Dodecagon (3 of 4)1/7/ :41:31 PM

50 Practice with Sum of Interior Angles Murray Copyright Oswego City School District Regents Exam Prep Center (4 of 4)1/7/ :41:31 PM

51 Probability Involving AND and OR Math A Probability Involving AND and OR We saw the connective words AND and OR when we studied logic. Let's examine AND first: In logic, we learned that a sentence "p and q" is true only when both p and q are true. In probability, an outcome is in event "A and B" only when the outcome is in both event A and event B. Rule (for AND): n(a and B) means the number of outcomes in both A and B. n(s) means the total number of possible outcomes (1 of 3)1/7/ :37:14 PM

52 Probability Involving AND and OR Example: A die is rolled. What is the probability that the number is even and less than 4? Event A: Numbers on a die that are even: 2, 4, 6 Event B: Numbers on a die that are less than 4: 1, 2, 3 There is only one number (2) that is in both events A and B. Total outcomes S: Numbers on a die: 1, 2, 3, 4, 5, 6 (total = 6) Answer: Probability = 1/6 Let's examine OR: In logic, we learned that a sentence "p or q" is true when either (or both) p or q are true. In probability, an outcome is in event "A or B" when the outcome is in either (or both) event A or event B. Rule (for OR): The rule for OR takes into account those values that may get counted more than once when the probability is determined. Check out the example below. (2 of 3)1/7/ :37:14 PM

53 Probability Involving AND and OR Example: A die is rolled. What is the probability that the number is even or less than 4? Event A: Numbers on a die that are even: 2, 4, 6 P(A)=3/6 Event B: Numbers on a die that are less than 4: 1, 2, 3 P(B)=3/6 P(A and B) = 1/6 (see rule above) Answer: Probability = P(A) + P(B) - P(A and B) = 3/6 + 3/6-1/6 = 5/6 **Notice in this problem that the number 2 appears in both event A and event B. If we did not subtract the P(A and B), the answer would be 1 - which we know is not true since the number 5 appears in neither event. Roberts Copyright Oswego City School District Regents Exam Prep Center (3 of 3)1/7/ :37:14 PM

54 Practice with Probability Involving AND and OR Math A Working with Probability Involving AND & OR Answer the following questions dealing with probability. 1. A die is rolled. What is the probability that the number rolled is greater than 2 and even? Choose: 1/2 1/3 2/3 5/6 (1 of 5)1/7/ :41:14 PM

55 Practice with Probability Involving AND and OR 2. From a standard deck of cards, one card is drawn. What is the probability that the card is black and a jack? Choose: 1/2 1/13 1/26 1/52 3. A set of polygons contains a square, a rectangle, a rhombus and a trapezoid. If one polygon is chosen at random, what is the probability that the polygon has all sides congruent and all right angles? Choose: 1/4 1/3 1/2 3/4 (2 of 5)1/7/ :41:14 PM

56 Practice with Probability Involving AND and OR 4. A standard deck of cards is shuffled and one card is drawn. Find the probability that the card is a queen or an ace. Choose: 1/13 2/13 1/4 3/52 (3 of 5)1/7/ :41:14 PM

57 Practice with Probability Involving AND and OR 5. A standard deck of cards is shuffled and one card is drawn. Find the probability that the card is red or a jack. Choose: 7/52 1/13 7/13 1/26 6. A piggybank contains 2 quarters, 3 dimes, 4 nickels, and 5 pennies. One coin is removed at random. What is the probability that the coin is a dime or a nickel? Choose: 3/14 4/14 7/14 1/7 (4 of 5)1/7/ :41:14 PM

58 Practice with Probability Involving AND and OR Roberts Copyright Oswego City School District Regents Exam Prep Center (5 of 5)1/7/ :41:14 PM

59 Complement of an Event Complement of an Event Math A If A is an event within the sample space S of an activity or experiment, the complement of A (denoted A') consists of all outcomes in S that are not in A. The complement of A is everything else in the problem that is NOT in A. For each experiment, an event and its complement are shown: Experiment: Tossing a coin Event A The coin shows heads. Complement A' The coin shows tails. Experiment: Drawing a card Event A The card is black. Complement A' The card is red. The probability of the complement of an event is one minus the probability of the event. (1 of 3)1/7/ :31:04 PM

60 Complement of an Event P(A') = 1 - P(A) Example 1: A pair of dice are rolled. What is the probability of not rolling doubles? P(doubles) = 6/36 = 1/6 P(not doubles) = 1-1/6 = 5/6 Example 2: A pair of dice are rolled. What is the probability of rolling 10 or less? The complement of rolling "10 or less" is rolling 11 or 12. P(10 or less) = 1 - P(11 or 12) = 1 - [P(11) + P(12)] = 1 - (2/36 + 1/36) = 33/36 = 11/12 (2 of 3)1/7/ :31:04 PM

61 Complement of an Event Roberts Copyright Oswego City School District Regents Exam Prep Center (3 of 3)1/7/ :31:04 PM

62 Mutually Exclusive Events Math A Mutually Exclusive Events An event is a set of outcomes. It is a subset of the sample space for an activity or experiment. Event: Drawing a black card from a deck of standard cards. Probability of this event = 26/52 = 1/2 When an event corresponds to a single outcome of the activity, it is often called a simple event. Simple Event: Drawing the queen of spades from a deck of standard cards. Probability of this event = 1/52 (1 of 5)1/7/ :34:37 PM

63 Mutually Exclusive Events Two events that have NO outcomes in common are called mutually exclusive. These are events that cannot occur at the same time. Mutually exclusive - think of this as the 2 events together (mutually) agreeing to exclude (not include) each others' elements. They have agreed to be different - mutually exclusive. Example: A pair of dice is rolled. The events of rolling a 6 and of rolling a double have the outcome (3,3) in common. These two events are NOT mutually exclusive. A pair of dice is rolled. The events of rolling a 9 and of rolling a double have NO outcomes in common. These two events ARE mutually exclusive. For any two mutually exclusive events, the probability that an outcome will be in one event or the other event is the sum of their individual probabilities. (2 of 5)1/7/ :34:37 PM

64 Mutually Exclusive Events If A and B are mutually exclusive events, P(A or B) = P(A) + P(B) For any two events which are not mutually exclusive, the probability that an outcome will be in one event or the other event is the sum of their individual probabilities minus the probability of the outcome being in both events. Look out!! Don't get stuck on this one!!! If events A and B are NOT mutually exclusive, P(A or B) = P(A) + P(B) - P(A and B) Example 1: A pair of dice is rolled. What is the probability that the sum of the numbers rolled is either 7 or 11? Six outcomes have a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) P(7) = 6/36 (3 of 5)1/7/ :34:37 PM

65 Mutually Exclusive Events Two outcomes have a sum of 11: (5,6), (6,5) P(11) = 2/36 The sum of the numbers cannot be 7 and 11 at the same time, so these events are mutually exclusive. P(7 or 11) = P(7) + P(11) = 6/36 + 2/36 = 8/36 = 2/9 Example 2: A pair of dice is rolled. What is the probability that the sum of the numbers rolled is either an even number or a multiple of 3? Of the 36 possible outcomes, 18 are even sums. P(even) = 18/36 = 1/2 Sums of 3, 6, 9, and 12 are multiples of 3. There are 12 sums that are multiples of 3. P(multiple of 3)= 12/36 = 1/3 However, some of these outcomes appear in both events. The sums that are even and a multiple of 3 are 6 and 12. There are 6 ordered pairs with these sums. P(even AND a multiple of 3) = 6/36 = 1/6 P(even OR a multiple of 3) = 18/ /36-6/36 = 24/36 = 2/3 (4 of 5)1/7/ :34:37 PM

66 Mutually Exclusive Events Roberts Copyright Oswego City School District Regents Exam Prep Center (5 of 5)1/7/ :34:37 PM

67 Independent Events Independent Events Math A Two events are said to be independent if the result of the second event is not affected by the result of the first event. If A and B are independent events, the probability of both events occurring is the product of the probabilities of the individual events If A and B are independent events, P(A and B) = P(A) x P(B). Example: A drawer contains 3 red paperclips, 4 green paperclips, and 5 blue paperclips. One paperclip is taken from the drawer and then replaced. Another paperclip is taken from the drawer. What is the probability that the first paperclip is red and the second paperclip is blue? Because the first paper clip is replaced, the sample space of 12 paperclips does not change from the (1 of 3)1/7/ :33:24 PM

68 Practice with Events Math A Working with Events Answer the following questions dealing with mutually exclusive events, independent events, and complements of events. 1. A pair of dice is rolled. Two possible events are rolling a number greater than 8 and rolling an even number. Are these two events mutually exclusive events? See Sample Space for Dice Choose: yes no (1 of 8)1/7/ :40:48 PM

69 Practice with Events 2. A pair of dice is rolled. Two possible events are rolling a number less than 5 and rolling a number which is a multiple of 5. Are these two events mutually exclusive? Choose: yes no See Sample Space for Dice (2 of 8)1/7/ :40:48 PM

70 Practice with Events 3. A pair of dice is rolled. A possible event is rolling a multiple of 5. What is the probability of the complement of this event? See Sample Space for Dice Choose: 2/36 12/36 29/36 32/36 4. A pair of dice is rolled. Two possible events are rolling a number which is a multiple of 3 and rolling a number which is a multiple of 5. Are these two events mutually exclusive? Choose: yes no (3 of 8)1/7/ :40:48 PM

71 Practice with Events See Sample Space for Dice 5. A pair of dice is rolled and the resulting number is odd. Which of the following events is the complement of this event? 1. A number greater than 8 is rolled. 2. An even number is rolled. 3. A number less than 5 is rolled. 4. A multiple of 5 is rolled. Choose: (4 of 8)1/7/ :40:48 PM

72 Practice with Events See Sample Space for Dice 6. A paper bag contains 15 slips of paper. Eight of them are blue and are numbered from 1 to 8. Seven of them are pink and are numbered from 1 to 7. What is the probability of drawing a slip of paper with an even number? Choose: 8/15 7/15 6/15 5/15 (5 of 8)1/7/ :40:48 PM

73 Practice with Events 7. A paper bag contains 15 slips of paper. Eight of them are blue and are numbered from 1 to 8. Seven of them are pink and are numbered from 1 to 7. A slip is drawn from the bag and then replaced. A second slip is drawn. Are these two events independent? Choose: yes no 8. A paper bag contains 15 slips of paper. Eight of them are blue and are numbered from 1 to 8. Seven of them are pink and are numbered from 1 to 7. A possible event is that a blue slip is chosen. What is the probability of the complement of this event? Choose: 8/15 7/15 1/3 1/2 (6 of 8)1/7/ :40:48 PM

74 Practice with Events 9. A paper bag contains 15 slips of paper. Eight of them are blue and are numbered from 1 to 8. Seven of them are pink and are numbered from 1 to 7. You wish to draw a blue slip on the first draw and a pink slip on the second draw. The first slip is NOT replaced after being drawn. Are these events independent? Choose: yes no Roberts (7 of 8)1/7/ :40:48 PM

75 Independent Events first event to the second event. The events are independent. P(red then blue) = P(red) x P(blue) = 3/12 5/12 = 15/144 = 5/48. If the result of one event IS affected by the result of another event, the events are said to be dependent. If A and B are dependent events, the probability of both events occurring is the product of the probability of the first event and the probability of the second event once the first event has occurred. If A and B are dependent events, and A occurs first, P(A and B) = P(A) x P(B,once A has occurred) Example: A drawer contains 3 red paperclips, 4 green paperclips, and 5 blue paperclips. One paperclip is taken from the drawer and is NOT replaced. Another paperclip is taken from the drawer. What is the probability that the first paperclip is red and the second paperclip is blue? Because the first paper clip is NOT replaced, the sample space of the second event is changed. The sample space of the first event is 12 paperclips, but the sample space of the second event is now 11 paperclips. The events are dependent. P(red then blue) = P(red) x P(blue) = 3/12 5/11 = 15/132 = 5/44. (2 of 3)1/7/ :33:24 PM

76 Independent Events Don't panic! You can do it! Probability is mostly common sense. Roberts Copyright Oswego City School District Regents Exam Prep Center (3 of 3)1/7/ :33:24 PM

77 Practice with Events Math A Working with Events Answer the following questions dealing with mutually exclusive events, independent events, and complements of events. 1. A pair of dice is rolled. Two possible events are rolling a number greater than 8 and rolling an even number. Are these two events mutually exclusive events? See Sample Space for Dice Choose: yes no (1 of 8)1/7/ :40:48 PM

78 Practice with Events 2. A pair of dice is rolled. Two possible events are rolling a number less than 5 and rolling a number which is a multiple of 5. Are these two events mutually exclusive? Choose: yes no See Sample Space for Dice (2 of 8)1/7/ :40:48 PM

79 Practice with Events 3. A pair of dice is rolled. A possible event is rolling a multiple of 5. What is the probability of the complement of this event? See Sample Space for Dice Choose: 2/36 12/36 29/36 32/36 4. A pair of dice is rolled. Two possible events are rolling a number which is a multiple of 3 and rolling a number which is a multiple of 5. Are these two events mutually exclusive? Choose: yes no (3 of 8)1/7/ :40:48 PM

80 Practice with Events See Sample Space for Dice 5. A pair of dice is rolled and the resulting number is odd. Which of the following events is the complement of this event? 1. A number greater than 8 is rolled. 2. An even number is rolled. 3. A number less than 5 is rolled. 4. A multiple of 5 is rolled. Choose: (4 of 8)1/7/ :40:48 PM

81 Practice with Events See Sample Space for Dice 6. A paper bag contains 15 slips of paper. Eight of them are blue and are numbered from 1 to 8. Seven of them are pink and are numbered from 1 to 7. What is the probability of drawing a slip of paper with an even number? Choose: 8/15 7/15 6/15 5/15 (5 of 8)1/7/ :40:48 PM

82 Practice with Events 7. A paper bag contains 15 slips of paper. Eight of them are blue and are numbered from 1 to 8. Seven of them are pink and are numbered from 1 to 7. A slip is drawn from the bag and then replaced. A second slip is drawn. Are these two events independent? Choose: yes no 8. A paper bag contains 15 slips of paper. Eight of them are blue and are numbered from 1 to 8. Seven of them are pink and are numbered from 1 to 7. A possible event is that a blue slip is chosen. What is the probability of the complement of this event? Choose: 8/15 7/15 1/3 1/2 (6 of 8)1/7/ :40:48 PM

83 Practice with Events 9. A paper bag contains 15 slips of paper. Eight of them are blue and are numbered from 1 to 8. Seven of them are pink and are numbered from 1 to 7. You wish to draw a blue slip on the first draw and a pink slip on the second draw. The first slip is NOT replaced after being drawn. Are these events independent? Choose: yes no Roberts (7 of 8)1/7/ :40:48 PM