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Unit 7 - Probability

Unit 7 - Probability. Analytical Geometry. Bellringer 8/11/14. What do you know about Probability?. The Probability Song (T.D. Version). Today we’re going to learn about probabilities, and we’re going to make diagrams that look like trees.

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Unit 7 - Probability

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  1. Unit 7 - Probability Analytical Geometry

  2. Bellringer 8/11/14 What do you know about Probability?

  3. The Probability Song(T.D. Version) Today we’re going to learn about probabilities, and we’re going to make diagrams that look like trees. We’re going to flip some coins so we can draw our trees,  and we’re going to calculate our probabilities. Probability’s the likelihood of a chance event. Hey!! Ho!! Hey!!  It’s between 0 and 1 and that’s just common sense.  Hey!! Hey!!  An event is any outcome from an experiment. If you can figure them all out, then you’re intelligent. (Repeat)  Today we’re going to learn about probabilities, and we’re going to make diagrams that look like trees. We’re going to flip some coins so we can draw our trees,  and we’re going to calculate our probabilities.

  4. On the way to Probability…. Probability Event Outcome Experiment

  5. What is Probability? Probability is a measure of the likelihood of an event. It is the ratio of the number of ways a certain event can occur to the number of possible outcomes. 1. The probability of a given event is a fraction between 0 and 1. 2. The sum of all probabilities of every outcome should always equal 1. 3.

  6. What does it mean to have a probability of zero? A probability of zero means that an event is impossible.

  7. What does it mean to have a probability of One? Question: A single 6-sided die is rolled. What is the probability of rolling a number less than 7? Answer: Rolling a number less than 7 is a certain event since a single die has 6 sides, numbered 1 through 6. A probability of one means that an event is certainly going to happen.

  8. Why are probabilities between 0 and 1? Probabilities are always between 0 and 1 because every event has a chance of occurrence that lies between 0% (no chance of happening, corresponds to 0) and 100% (certainty of happening, corresponds to 1).

  9. Tree Diagrams Tree diagram: A tree-shaped diagram that illustrates sequentially the possible outcomes of a given event. Here is a tree diagram for the toss of a coin:

  10. Create a tree diagram for the possible outcomes and probabilities for the two tosses. Probabilities Tree Diagram The Sum of all Possible outcomes

  11. Bellringer 8/12/14 Kendra is playing a card game with a standard 52-card deck. She wants her first draw to be a heart or an ace. How many ways can Kendra draw a heart or an ace? How many ways can Kendra draw a card that is neither a heart nor an ace?

  12. IntroductionProbability is a number from 0 to 1 inclusive or a percent from 0% to 100% inclusive that indicates how likely an event is to occur. In the study of probability, an experiment is any process or action that has observable results. The results are called outcomes. The set of all possible outcomes is called the sample space. An event is an outcome or set of outcomes of an experiment; therefore, an event is a subset of the sample space, meaning a set whose elements are in another set, the sample space. In this lesson, you will learn to describe events as subsets of the sample space. For example, drawing a card from a deck of cards is an experiment. All the cards in the deck are possible outcomes, and they comprise the sample space. The event that a jack of hearts is drawn is a single outcome; it is a set with one element: {jack of hearts}. An element is an item or a member in a set. The event that a red number card less than 5 is drawn is a set with eight elements: {ace of hearts, ace of diamonds, 2 of hearts, 2 of diamonds, 3 of hearts, 3 of diamonds, 4 of hearts, 4 of diamonds}. Each event is a subset of the sample space.Note that the words “experiment” and “event” in probability have very specific meanings that are not always the same as the everyday meanings. In the example on the previous slide, “drawing a card” describes an experiment, not an event. “Drawing a jack of hearts” and “drawing a red number card less than 5” describe events, and those events are sets of outcomes.The important distinction to remember is that an event is a set of one or more outcomes, while the action or process is the experiment.

  13. Bellringer 8/13/14 What is the Sample Space of rolling 2 dice? Write an event for the experiment mentioned in Question1.

  14. Key ConceptsA set is a list or collection of items. Set A is a subset of set B, denoted by A ⊂ B, if all the elements of A are also in B. For example, if B = {1, 2, 3, x}, then some subsets of B are {1, 2, 3}, {2, x}, {3}, and {1, 2, 3, x}. Note that a set is a subset of itself.An empty set is a set that has no elements. An empty set is denoted by . An empty set is also called a null set. Equal sets are sets with all the same elements. For example, consider sets A, B, and C as follows: A is the set of integers between 0 and 6 that are not odd. B is the set of even positive factors of 4. C = {2, 4}A = B = C because they all have the same elements, 2 and 4.The union of sets A and B, denoted by , is the set of elements that are in either A or B or both A and B.

  15. Bellringer 8/14/14 What is the probability of rolling an even number on a 6-sided die? For the previous question write the set for the Sample space and the subset for the event.

  16. Common Errors/Misconceptions1) confusing the meanings of event and experiment2) confusing union and intersection of sets 3) neglecting order, thereby neglecting to identify different outcomes such as HT and TH

  17. Key Concepts, continuedAn intersection is a set whose elements are each in both of two other sets. The intersection of sets A and B, denoted by , is the set of elements that are in both A and B.For example, if A = {a, b, c} and B = {m, a, t, h}, then:An event is an outcome or set of outcomes, so an event is a subset of the sample space. The complement of set A, denoted by , is the set of elements that are in some universal set, but not in A. The complement of A is the event that does not occur.Sometimes it is helpful to draw tables or diagrams to visualize outcomes and the relationships between events. A Venn diagram is a diagram that shows how two or more sets in a universal set are related. In this diagram, members in event A also fit the criteria for members in event B, so the circles overlap:

  18. Bellringer 8/18/14 1) What is the Probability of your phone landing screen side down when you drop it? 2) How do you express Probability as a ratio? **Turn in your Take home test to the appropriate current work folder for your class** 1st – Green, 2nd – Gray, 7th - Blue

  19. Probability words to know! Probability: a measure of the likelihood of an event. It is the ratio of the number of ways a certain event can occur to the number of possible outcomes. The probability of a given event is a fractionbetween 0and 1. The sum of all probabilities of every outcome should always equal 1. • Experiment: a process or ACTION with an observable result. • Outcomes: the observable results of an experiment. • Sample Space: the set of ALL possible outcomes of an experiment. • Event: an outcome or set of outcomes of an experiment. • Subset: a set whose elements are all in another set.

  20. Need to know about Sets! • Set: a list or collection of items • Subset – denoted by ⊂ , as in A ⊂ B where A is a subset of B, where all of the elements of A are also in B • Empty set: also known as a null set, is a set with no elements denoted by • Equal sets: sets with the exact same elements • Union of Sets: joining of all of the elements of two sets denoted by , where A and B are being combined • Intersection of sets: the elements that are alike or shared by two sets denoted by , the intersection of A and B contains the elements that are in both A and B • Complement of a set: the set of elements that are in some universal set, but not in set A, denoted by Ā or A’

  21. Common Errors/Misconceptions • confusing the meanings of event and experiment • confusing union and intersection of sets • neglecting order, thereby neglecting to identify different outcomes such as HT and TH

  22. Guided Practice Example 1 Hector has entered the following names in the contact list of his new cell phone: Alicia, Brisa, Steve, Don, and Ellis. He chooses one of the names at random to call. Consider the following events. B: The name begins with a vowel. E: The name ends with a vowel. Draw a Venn diagram to show the sample space and the events B and E. Then describe each of the following events by listing outcomes. B E

  23. Guided Practice: Example 1, continued Draw a Venn diagram. Use a rectangle for the sample space. Use circles or elliptical shapes for the events B and E. Write the students’ names in the appropriate sections to show what events they are in. 7.1.1: Describing Events

  24. Guided Practice: Example 1, continued List the outcomes of B. B = {Ellis, Alicia} 7.1.1: Describing Events

  25. Guided Practice: Example 1, continued List the outcomes of E. E = {Alicia, Brisa, Steve} 7.1.1: Describing Events

  26. Guided Practice: Example 1, continued List the outcomes of B = {Ellis, Alicia} E = {Alicia, Brisa, Steve} is the intersection of events B and E. Identify the outcome(s) common to both events. = {Alicia} 7.1.1: Describing Events

  27. Guided Practice: Example 1, continued List the outcomes of B = {Ellis, Alicia} E = {Alicia, Brisa, Steve} is the union of events B and E. Identify the outcomes that appear in either event or both events. 7.1.1: Describing Events

  28. Guided Practice: Example 1, continued List the outcomes of . B = {Ellis, Alicia} Sample space = {Alicia, Brisa, Steve, Don, Ellis} is the set of all outcomes that are in the sample space, but not in B. = {Brisa, Steve, Don} 7.1.1: Describing Events

  29. Guided Practice: Example 1, continued List the outcomes of . B = {Ellis, Alicia} E = {Alicia, Brisa, Steve} = {Ellis, Alicia, Brisa, Steve} Sample space = {Alicia, Brisa, Steve, Don, Ellis} is the set of all outcomes that are in the sample space, but not in . = {Don} ✔ 7.1.1: Describing Events

  30. Guided Practice Example 2 An experiment consists of rolling a pair of dice. How many ways can you roll the dice so that the product of the two numbers rolled is less than their sum?

  31. Guided Practice: Example 2, continued Begin by showing the sample space. This diagram of ordered pairs shows the sample space. Key: (2, 3) means 2 on the first die and 3 on the second die. 7.1.1: Describing Events

  32. Guided Practice: Example 2, continued Identify all the outcomes in the event “the product is less than the sum.” For (1, 1), the product is 1 × 1 = 1 and the sum is 1 + 1 = 2, so the product is less than the sum. For (1, 2), the product is 1 × 2 = 2 and the sum is 1 + 2 = 3, so the product is less than the sum. The tables on the next two slides show all the possible products and sums. 7.1.1: Describing Events

  33. Guided Practice: Example 2, continued 7.1.1: Describing Events

  34. Guided Practice: Example 2, continued 7.1.1: Describing Events

  35. Guided Practice: Example 2, continued By checking all the outcomes in the sample space, you can verify that the product is less than the sum for only these outcomes: 7.1.1: Describing Events

  36. Guided Practice: Example 2, continued Count the outcomes that meet the event criteria. There are 11 ways to roll two dice so that the product is less than the sum. ✔ 7.1.1: Describing Events

  37. Bellringer 8/19/14 Create an experiment with 2 events. Draw a Venn Diagram to illustrate the experiment with the two events. Put away your notebooks and other items. I should only see a pencil and eraser on your desk!

  38. Key Concepts The conditional probability of B given A is the probability that event B occurs, given that event A has already occurred. If A and B are two events from a sample space with P(A) ≠ 0, then the conditional probability of B given A, denoted , has two equivalent expressions: The second formula can be rewritten as ,is read “the probability of B given A.” Using set notation, conditional probability is written like this: The “conditional probability of B given A” only has meaning if event A has occurred. That is why the formula forhas the requirement that P(A)≠0. The conditional probability formula can be solved to obtain a formula for P(A and B), as shown on the next slide.

  39. Key Concepts, continued Remember that independent events are two events such that the probability of both events occurring is equal to the product of the individual probabilities. Two events A and B are independent if and only if P(A and B) = P(A) • P(B). Using set notation, . The occurrence or non-occurrence of one event has no effect on the probability of the other event. If A and B are independent, then the formula for P(A and B) is the equation used in the definition of independent events, as shown below.

  40. Key Concepts, continued The following statements are equivalent. In other words, if any one of them is true, then the others are all true. Events A and B are independent. The occurrence of A has no effect on the probability of B; that is, The occurrence of B has no effect on the probability of A; that is, P(A and B) = P(A) • P(B). Note: For real-world data, these modified tests for independence are sometimes used: Events A and B are independent if the occurrence of A has no significant effect on the probability of B; that is, Events A and B are independent if the occurrence of B has no significant effect on the probability of A; that is, When using these modified tests, good judgment must be used when deciding whether the probabilities are close enough to conclude that the events are independent.

  41. Bellringer 8/20/14 Define Probability Define Experiment Define Event Define Outcome

  42. Guided Practice Example 1 Alexis rolls a pair of number cubes. What is the probability that both numbers are odd if their sum is 6? Interpret your answer in terms of a uniform probability model.

  43. Guided Practice: Example 1, continued Assign variable names to the events and state what you need to find, using conditional probability. Let A be the event “Both numbers are odd.” Let B be the event “The sum of the numbers is 6.” You need to find the probability of A given B. That is, you need to find 7.2.1: Introducing Conditional Probability

  44. Guided Practice: Example 1, continued Show the sample space. (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) Key: (2, 3) means 2 on the first cube and 3 on the second cube. 7.2.1: Introducing Conditional Probability

  45. Guided Practice: Example 1, continued Identify the outcomes in the events. The outcomes for A are bold and purple. A: Both numbers are odd. (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) 7.2.1: Introducing Conditional Probability

  46. Guided Practice: Example 1, continued The outcomes for B are bold and purple. B: The sum of the numbers is 6. (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) 7.2.1: Introducing Conditional Probability

  47. Guided Practice: Example 1, continued Identify the outcomes in the events and B. Use the conditional probability formula: = the outcomes that are in A and also in B. B = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)} 7.2.1: Introducing Conditional Probability

  48. Guided Practice: Example 1, continued Find and has 3 outcomes; the sample space has 36 outcomes. Bhas 5 outcomes; the sample space has 36 outcomes. 7.2.1: Introducing Conditional Probability

  49. Guided Practice: Example 1, continued Find 7.2.1: Introducing Conditional Probability

  50. Guided Practice: Example 1, continued 7.2.1: Introducing Conditional Probability

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