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Being Equally Likely

Being Equally Likely. Ellie Qu. Audience. This presentation is prepared for high school students who are learning Probability and Chance. Some of the material can also be useful for students who are taking Statisitcs . . Objective.

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Being Equally Likely

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  1. Being Equally Likely Ellie Qu

  2. Audience This presentation is prepared for high school students who are learning Probability and Chance. Some of the material can also be useful for students who are taking Statisitcs.

  3. Objective The purpose of this study guide is to review the first Chapter of Probability and Chance. This presentation is useful to go through before a test or final exam. After completing this guide. You should be not only recall the materials covered in class, but also know how to solve typical problems. Now, please take your text book and notes out, let’s go through this presentation together!

  4. Home One event, all outcomes equally likely Games • What is a venn diagram? • Probability Rules • Quiz • What is sample space? Step One

  5. Example: For instance, when you roll a pair of dice, you might ask how likely you are to roll a seven. Then you are asking what is the probability to roll a seven. What is Probability ? Note: The probability of an occurance of an event can be expressed as a fraction or a decimal between 0 and 1. Definition: The study of probability helps us figure out the likelihood of something happening. What is Sample Space?

  6. Example: For example, if the probability of picking a redmarble from a jar that contains 4 red marblesand 6 blue marbles is 4/10 or 2/5, then the probability of not picking a red marble is equalto 1 - 4/10 = 6/10 or 3/5, which is also the probability of picking a blue marble. That is,given this example, the probability of picking a red marble plus the probability of picking ablue marble will equal 1 (or 100 percent). In this case, 10 is the sample space. What is Sample Space ? Definition: The sample space is a set consisting of all the possible outcomes of an event. Equally Likely Outcomes

  7. Example: Suppose we have ten marbles, 4 red and 6 blue. What is the probability if we picked a red marble without looking? (4/10) What is the probability if we picked a blue marble without looking? (6/10) Conclusion: The probability of picking any red marble is equally likely, and the probability of picking any blue is equally likely. equally likely outcomes Definition: Any individual has the same chance of being chosen. Probability Rules

  8. Rule 2: The sum of all the probabilities in the sample space is 1. Rule 3: The probability of an event which cannot occur is 0. Rule 4: The probability of an event not occurring is one minus the probability of it occurring: P(E') = 1 - P(E) What is Probability Rules? Rule 1: All probabilities are between 0 and 1 inclusive: 0 <= P(E) <= 1 What is a Venn Diagram

  9. Example: Below is a venn diagram. A and B are events in a sample space, S. The intersection of A and B is written as : P(A and B). The union of A and B is written as : P(A or B). What is A Venn Diagram? Definition: A Venn Diagram describes the probabilities of different events graphically. Examples of Venn Diagram

  10. P(A and B) = P(A) . P(B) P(A) = 1/2 P(B) = 1/2P(A and B) = 1/2 . 1/2 = 1/4 Examples using a vein diagram • Given: Suppose a high school consists of 25% juniors,15% seniors, and the remaining 60% is students of other grades. The relative frequency of students who are either juniors and seniors is 40%. Given: What is the probability that two tails occurs when two coins are tossed? Let A represent the occurrence of a tail on the first coin and B represent the occurrence of a tail on the second coin. P(A or B) = P(A) + P(B) P(J or S) = 0.25 + 0.15 which equals 0.40 Application problems

  11. Application Problems • #1 What is the probability of drawing one ace and then another ace from a deck of playing cards? Next problem Answer Hint

  12. Application Problems • #2 Suppose we have two dice. A is the event that 4 shows on the first die, and B is the event that 4 shows on the second die. If both dice are rolled at once, what is the probability that two 4s occur? Previous Problem Hint Answer

  13. Hint: First die: 1/6 second die: 1/6 Go back to problem

  14. P(A) = 1/6P(B) = 1/6P(A and B) = P(A) . P(B) = 1/6 . 1/6 = 1/36 Answer: Go back to problem Go to Quiz

  15. Since there are 4 aces in a 52 deck of cards, the probability of drawing one ace is 4/52. After the first draw, the 51 cards remaining contain 3 aces and therefore the probability of drawing an ace on the second draw is 3/51. Hint: Go back to problem

  16. Answer: 4/52 . 3/51 = 1/221 Go back to problem Go to the next problem

  17. Quiz Time! • #1: You pick two cards out of a standard pack of 52. the fist card is a king. What’s the probability of the second card also being a king? A) 1/13 B) 1/17 C) 3/13 D) 1/14

  18. A) 1/13 Incorrect There are 51 cards left. Three of those are a king, so you have three chances out of fifty-one, which equals 1/17. B is the correct answer. Go back to problem Go to the next problem

  19. B) 1/17 Correct! There are 51 cards left. Three of those are a king, so you have three chances out of fifty-one, which equals 1/17. B is the correct answer. Go back to problem Go to the next problem

  20. C) 3/13 Incorrect There are 51 cards left. Three of those are a king, so you have three chances out of fifty-one, which equals 1/17. B is the correct answer. Go back to problem Go to the next problem

  21. D) 1/14 Incorrect There are 51 cards left. Three of those are a king, so you have three chances out of fifty-one, which equals 1/17. B is the correct answer. Go back to problem Go to the next problem

  22. Quiz Time! • #2: If P(R and B) = 0.17, P(R)=0.6, and P(B)=0.3, calculate P(R or B) A) 0.73 B) 0.9 C) 0.17 D) 0.5 previous problem

  23. A) 0.73 Correct! P(R or B) = P(R)+P(B)-P(R and B) = 0.6 + 0.3 – 0.17 = 0.73 Go back to problem Go to the next problem

  24. B) 0.9 Incorrect! P(R or B) = P(R)+P(B)-P(R and B) = 0.6 + 0.3 – 0.17 = 0.73 A is the correct answer. Go back to problem Go to the next problem

  25. C) 0.17 Incorrect P(R or B) = P(R)+P(B)-P(R and B) = 0.6 + 0.3 – 0.17 = 0.73 A is the correct answer. Go back to problem Go to the next problem

  26. D) 0.5 Incorrect P(R or B) = P(R)+P(B)-P(R and B) = 0.6 + 0.3 – 0.17 = 0.73 A is the correct answer. Go back to problem Go to the next problem

  27. Quiz Time! • #3: A large basket of fruit contains 3 oranges. 2 apples and 5 bananas. If a piece of fruit is chosen at random, what is the probability of getting an orange or a banana? A) 1/10 B) 1/2 C) 7/10 D) 4/5 previous problem

  28. A) 1/10 Incorrect! P( orange U banana)= 3/10 + 5/10 = 8/10 = 4/5 D is the correct answer. Go back to problem Go to the next problem

  29. B) 1/2 Incorrect! P( orange U banana)= 3/10 + 5/10 = 8/10 = 4/5 D is the correct answer. Go back to problem Go to the next problem

  30. C) 7/10 Incorrect P( orange U banana)= 3/10 + 5/10 = 8/10 = 4/5 D is the correct answer. Go back to problem Go to the next problem

  31. D) 4/5 Correct! P( orange U banana)= 3/10 + 5/10 = 8/10 = 4/5 D is the correct answer. Go back to problem Go to the next problem

  32. Quiz Time! • #4: A pair of dice is rolled. What’s the probability of getting a sum of 2? A) 1/6 B) 1/3 C) 1/36 D) 1/64 previous problem

  33. A) 1/6 Incorrect! 1 2 3 4 5 6 1 2 3 4 5 6 Possible Outcomes of sums: 36. Only 1+1=2, so the probability is 1/36. C is the right answer. Go back to problem Go to the next problem

  34. B) 1/3 Incorrect! 1 2 3 4 5 6 1 2 3 4 5 6 Possible Outcomes of sums: 36. Only 1+1=2, so the probability is 1/36. C is the right answer. Go back to problem Go to the next problem

  35. C) 1/36 Correct 1 2 3 4 5 6 1 2 3 4 5 6 Possible Outcomes of sums: 36. Only 1+1=2, so the probability is 1/36. C is the right answer. Go back to problem Go to the next problem

  36. D) 1/64 Incorrect 1 2 3 4 5 6 1 2 3 4 5 6 Possible Outcomes of sums: 36. Only 1+1=2, so the probability is 1/36. C is the right answer. Go back to problem Go to the next problem

  37. Quiz Time! • #5: When two events are mutually exclusive, they must be independent. True False previous problem

  38. A) True Correct! Two events are mutually exclusive, they must be independent; however, it they are independent, they might not be mutually exclusive. Go back to problem Go to Games

  39. A) False Incorrect! Two events are mutually exclusive, they must be independent; however, it they are independent, they might not be mutually exclusive. Go back to problem Go to Games

  40. Links to Probability Games Coin Flip, Dice Roll: http://www.betweenwaters.com/probab/probab.html Application games: http://www.bbc.co.uk/skillswise/numbers/handlingdata/probability/game.shtml MathHelp Notebook on Probability: http://www.ucl.ac.uk/Mathematics/geomath/level2/prob/MHpb.html

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