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Pledge & Moment of Silence

Pledge & Moment of Silence. . Pre-Algebra HOMEWORK. Page 449 #1-8 & Page 453 #1-6. Our Learning Goal.

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Pledge & Moment of Silence

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  1. Pledge & Moment of Silence

  2. Pre-Algebra HOMEWORK Page 449 #1-8 & Page 453 #1-6

  3. Our Learning Goal Students will be able to find theoretical probabilities, including dependent and independent events; estimate probabilities using experiments and simulations; use The Fundamental Counting Principle, permutations, and combinations; and convert between probability and odds of a specified outcome.

  4. Our Learning Goal Assignments • Learn to find he probability of an event by using the definition of probability (9-1) • Learn to estimate probability using experimental methods (9-2)

  5. Student Learning Goal Chart

  6. FAST TRACK! 9-1 and 9-2

  7. Today’s Learning Goal Assignment Learn to find the probability of an event by using the definition of probability.

  8. Lesson Quiz Use the table to find the probability of each event. 1. 1 or 2 occurring 2. 3 not occurring 3. 2, 3, or 4 occurring 0.351 0.874 0.794

  9. 9-2 Experimental Probability Today’s Learning Goal Assignment Learn to estimate probability using experimental methods.

  10. 9-2 Experimental Probability Lesson Quiz: Part 1 1. Of 425, 234 seniors were enrolled in a math course. Estimate the probability that a randomly selected senior is enrolled in a math course. 2. Mason made a hit 34 out of his last 125 times at bat. Estimate the probability that he will make a hit his next time at bat. 0.55, or 55% 0.27, or 27%

  11. 9-2 Experimental Probability Lesson Quiz: Part 2 3. Christina polled 176 students about their favorite ice cream flavor. 63 students’ favorite flavor is vanilla and 40 students’ favorite flavor is strawberry. Compare the probability of a student’s liking vanilla to a student’s liking strawberry. about 36% to about 23%

  12. Vocabulary experiment trial outcome sample space event probability impossible certain

  13. An experiment is an activity in which results are observed. Each observation is called a trial, and each result is called an outcome. The sample space is the set of all possible outcomes of an experiment. Experiment Sample Space flipping a coin heads, tails rolling a number cube 1, 2, 3, 4, 5, 6 guessing the number of whole numbers jelly beans in a jar

  14. An event is any set of one or more outcomes. The probability of an event, written P(event), is a number from 0 (or 0%) to 1 (or 100%) that tells you how likely the event is to happen. • A probability of 0 means the event is impossible, or can never happen. • A probability of 1 means the event is certain, or has to happen. • The probabilities of all the outcomes in the sample space add up to 1.

  15. 1 1 3 4 2 4 Never Happens about Always happens half the time happens 1 0 0 0.25 0.5 0.75 1 0% 25% 50% 75% 100%

  16. Additional Example 1A: Finding Probabilities of Outcomes in a Sample Space Give the probability for each outcome. A.The basketball team has a 70% chance of winning. The probability of winning is P(win) = 70% = 0.7. The probabilities must add to 1, so the probability of not winning is P(lose) = 1 – 0.7 = 0.3, or 30%.

  17. Try This: Example 1A Give the probability for each outcome. A. The polo team has a 50% chance of winning. The probability of winning is P(win) = 50% = 0.5. The probabilities must add to 1, so the probability of not winning is P(lose) = 1 – 0.5 = 0.5, or 50%.

  18. 3 8 Three of the eight sections of the spinner are labeled 1, so a reasonable estimate of the probability that the spinner will land on 1 is P(1) = . Additional Example 1B: Finding Probabilities of Outcomes in a Sample Space Give the probability for each outcome. B.

  19. Three of the eight sections of the spinner are labeled 2, so a reasonable estimate of the probability that the spinner will land on 2 is P(2) = . 3 2 1 2 3 3 8 8 8 8 8 4 Two of the eight sections of the spinner are labeled 3, so a reasonable estimate of the probability that the spinner will land on 3 is P(3) = = . + + = 1 Additional Example 1B Continued Check The probabilities of all the outcomes must add to 1. 

  20. 1 1 6 6 One of the six sides of a cube is labeled 1, so a reasonable estimate of the probability that the spinner will land on 1 is P(1) = . One of the six sides of a cube is labeled 2, so a reasonable estimate of the probability that the spinner will land on 1 is P(2) = . Try This: Example 1B Give the probability for each outcome. B. Rolling a number cube.

  21. 1 1 1 6 6 6 One of the six sides of a cube is labeled 3, so a reasonable estimate of the probability that the spinner will land on 1 is P(3) = . One of the six sides of a cube is labeled 4, so a reasonable estimate of the probability that the spinner will land on 1 is P(4) = . One of the six sides of a cube is labeled 5, so a reasonable estimate of the probability that the spinner will land on 1 is P(5) = . Try This: Example 1B Continued

  22. 1 1 1 1 1 1 6 6 6 6 6 6 One of the six sides of a cube is labeled 6, so a reasonable estimate of the probability that the spinner will land on 1 is P(6) = . 1 + + + + + = 1 6 Try This: Example 1B Continued Check The probabilities of all the outcomes must add to 1. 

  23. To find the probability of an event, add the probabilities of all the outcomes included in the event.

  24. Additional Example 2A: Finding Probabilities of Events A quiz contains 5 true or false questions. Suppose you guess randomly on every question. The table below gives the probability of each score. A. What is the probability of not guessing 3 or more correct? The event “not three or more correct” consists of the outcomes 0, 1, and 2. P(not 3 or more) = 0.031 + 0.156 + 0.313 = 0.5, or 50%.

  25. Try This: Example 2A A quiz contains 5 true or false questions. Suppose you guess randomly on every question. The table below gives the probability of each score. A. What is the probability of guessing 3 or more correct? The event “three or more correct” consists of the outcomes 3, 4, and 5. P(3 or more) = 0.313 + 0.156 + 0.031 = 0.5, or 50%.

  26. Additional Example 2B: Finding Probabilities of Events A quiz contains 5 true or false questions. Suppose you guess randomly on every question. The table below gives the probability of each score. B. What is the probability of guessing between 2 and 5? The event “between 2 and 5” consists of the outcomes 3 and 4. P(between 2 and 5) = 0.313 + 0.156 = 0.469, or 46.9%

  27. Try This: Example 2B A quiz contains 5 true or false questions. Suppose you guess randomly on every question. The table below gives the probability of each score. B. What is the probability of guessing fewer than 3 correct? The event “fewer than 3” consists of the outcomes 0, 1, and 2. P(fewer than 3) = 0.031 + 0.156 + 0.313 = 0.5, or 50%

  28. Additional Example 2C: Finding Probabilities of Events A quiz contains 5 true or false questions. Suppose you guess randomly on every question. The table below gives the probability of each score. C. What is the probability of guessing an even number of questions correctly (not counting zero)? The event “even number correct” consists of the outcomes 2 and 4. P(even number correct) = 0.313 + 0.156 = 0.469, or 46.9%

  29. Try This: Example 2C A quiz contains 5 true or false questions. Suppose you guess randomly on every question. The table below gives the probability of each score. C. What is the probability of passing the quiz (getting 4 or 5 correct) by guessing? The event “passing the quiz” consists of the outcomes 4 and 5. P(passing the quiz) = 0.156 + 0.031 = 0.187, or 18.7%

  30. Six students are in a race. Ken’s probability of winning is 0.2. Lee is twice as likely to win as Ken. Roy is as likely to win as Lee. Tracy, James, and Kadeem all have the same chance of winning. Create a table of probabilities for the sample space. 14 Additional Example 3: Problem Solving Application

  31. 1 1 1 4 4 • P(Roy) = P(Lee) =  0.4 = 0.1 Understand the Problem Additional Example 3 Continued The answer will be a table of probabilities. Each probability will be a number from 0 to 1. The probabilities of all outcomes add to 1. List the important information: • P(Ken) = 0.2 • P(Lee) = 2  P(Ken) = 2  0.2 = 0.4 • P(Tracy) = P(James) = P(Kadeem)

  32. Make a Plan 2 Additional Example 3 Continued You know the probabilities add to 1, so use the strategy write an equation. Let p represent the probability for Tracy, James, and Kadeem. P(Ken) + P(Lee) + P(Roy) + P(Tracy) + P(James) + P(Kadeem) = 1 0.2 + 0.4 + 0.1 + p + p + p = 1 0.7 + 3p = 1

  33. 3 Solve 3p 0.3 3 3 = Divide both sides by 3. Additional Example 3 Continued 0.7 + 3p = 1 –0.7 –0.7Subtract 0.7 from both sides. 3p = 0.3 p = 0.1

  34. 4 Look Back Additional Example 3 Continued Check that the probabilities add to 1. 0.2 + 0.4 + 0.1 + 0.1 + 0.1 + 0.1 = 1 

  35. Try This: Example 3 Four students are in the Spelling Bee. Fred’s probability of winning is 0.6. Willa’s chances are one-third of Fred’s. Betty’s and Barrie’s chances are the same. Create a table of probabilities for the sample space.

  36. 1 1 1 3 3 • P(Willa) =  P(Fred) =  0.6 = 0.2 Understand the Problem Try This: Example 3 Continued The answer will be a table of probabilities. Each probability will be a number from 0 to 1. The probabilities of all outcomes add to 1. List the important information: • P(Fred) = 0.6 • P(Betty) = P(Barrie)

  37. Make a Plan 2 Try This: Example 3 Continued You know the probabilities add to 1, so use the strategy write an equation. Let p represent the probability for Betty and Barrie. P(Fred) + P(Willa) + P(Betty) + P(Barrie) = 1 0.6 + 0.2 + p + p = 1 0.8 + 2p = 1

  38. 3 Solve Try This: Example 3 Continued 0.8 + 2p = 1 –0.8 –0.8Subtract 0.8 from both sides. 2p = 0.2 p = 0.1

  39. 4 Look Back Try This: Example 3 Continued Check that the probabilities add to 1. 0.6 + 0.2 + 0.1 + 0.1 = 1 

  40. Lesson Quiz Use the table to find the probability of each event. 1. 1 or 2 occurring 2. 3 not occurring 3. 2, 3, or 4 occurring 0.351 0.874 0.794

  41. 9-2 Experimental Probability Warm Up Problem of the Day Lesson Presentation Pre-Algebra

  42. 9-2 Experimental Probability Pre-Algebra Warm Up Use the table to find the probability of each event. 1.A or B occurring 2. C not occurring 3. A, D, or E occurring 0.494 0.742 0.588

  43. Problem of the Day A spinner has 4 colors: red, blue, yellow, and green. The green and yellow sections are equal in size. If the probability of not spinning red or blue is 40%, what is the probability of spinning green? 20%

  44. Today’s Learning Goal Assignment Learn to estimate probability using experimental methods.

  45. Vocabulary experimental probability

  46. In experimental probability, the likelihood of an event is estimated by repeating an experiment many times and observing the number of times the event happens. That number is divided by the total number of trials. The more the experiment is repeated, the more accurate the estimate is likely to be. number of times the event occurs total number of trials probability 

  47. 186 number of spins that landed on 2 = total number of spins 500 Additional Example 1A: Estimating the Probability of an Event A. The table shows the results of 500 spins of a spinner. Estimate the probability of the spinner landing on 2. probability  The probability of landing on 2 is about 0.372, or 37.2%.

  48. 523 1000 Try This: Example 1A A. Jeff tosses a quarter 1000 times and finds that it lands heads 523 times. What is the probability that the next toss will land heads? Tails? = 0.523 P(heads) = P(heads) + P(tails) = 1 The probabilities must equal 1. 0.523 + P(tails) = 1 P(tails) = 0.477

  49. number of Canadian license plates 21 = total number of license plates 60 Additional Example 1B: Estimating the Probability of an Event B. A customs officer at the New York–Canada border noticed that of the 60 cars that he saw, 28 had New York license plates, 21 had Canadian license plates, and 11 had other license plates. Estimate the probability that a car will have Canadian license plates. probability  = 0.35 The probability that a car will have Canadian license plates is about 0.35, or 35%.

  50. number of plasma displays 13 = = total number of TVs 13 + 37 1350 Try This: Example 1B B. Josie sells TVs. On Monday she sold 13 plasma displays and 37 tube TVs. What is the probability that the first TV sold on Tuesday will be a plasma display? A tube TV? probability ≈ P(plasma) = 0.26 P(plasma) + P(tube) = 1 0.26 + P(tube) = 1 P(tube) = 0.74

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