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Introduction to Simulations and Experimental Probability

Introduction to Simulations and Experimental Probability. Introduction to Probability. Conditions for a “fair game”. a game is fair if… all players have an equal chance of winning, or each player can expect to win or lose the same number of times in the long run.

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Introduction to Simulations and Experimental Probability

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  1. Introduction to Simulations and Experimental Probability Introduction to Probability

  2. Conditions fora “fair game” • a game is fair if… • all players have an equal chance of winning, or • each player can expect to win or lose the same number of times in the long run http://mathdemos.gcsu.edu/plinko/ http://probability.ca/jeff/java/utday/ http://math.dartmouth.edu/~dlittle/java/Plinko/

  3. Important vocabulary • a trial is one repetition of an experiment • random variable: a variable whose value corresponds to the outcome of a random event

  4. Vocabulary / Terminology • expected value: (informally) the value to which the average of the random variable’s values tends after many repetitions; also called the “mean” • event: a set of possible outcomes of an experiment • simulation: an experiment that models an actual event

  5. A Definition of Probability • A measure of the likelihood of an event is called the probability of the event. • It is based on how often a particular event occurs in comparison with the total number of trials. • Probabilities derived from experiments are known as experimental probabilities.

  6. Experimental Probability • Experimental probability is the observed probability (also known as the relative frequency) of an event, A, in an experiment. • It is found using the following formula: P(A) = number of times A occurs total number of trials Note: probability is expressed as a number between 0 and 1

  7. Exercises and Assignment • The Coffee Game is an example of a simulation (an experiment that models an actual event). • In pairs, work through Investigation 1 (pp. 203-204) • Complete the 3 discussion questions on p. 204 • Read through Example 2 -- solution 1, p. 207 • Homework: p. 209, #2 (omit 2b i) and p. 211, #9

  8. Theoretical Probability Chapter 4.2 – An Introduction to Probability Mathematics of Data Management (Nelson) MDM 4U Author: Gary Greer and James Gauthier (with K. Myers)

  9. Gerolamo Cardano • Born in 1501, Pavia, Duchy of Milan (today, part of Italy) • Died: 1571 in Rome • Physician, inventor, mathematician, chess player, gambler

  10. Games of Chance • Most historians agree that the modern study of probability began with Gerolamo Cardano’s analysis of “Games of Chance” in the 1500s. • http://encyclopedia.thefreedictionary.com /Gerolamo Cardano • http://www-gap.dcs.st-and.ac.uk/~history /Mathematicians/Cardan.html

  11. A few terms… • simple event: an event that consists of exactly one outcome • sample space: the collection of all possible outcomes of the experiment • event space: the collection of all outcomes of an experiment that correspond to a particular event

  12. General Definition of Probability • assuming that all outcomes are equally likely, the probability of event A is: P(A) = n(A) n(S) • where n(A) is the number of elements in the event space and n(S) is the number of elements in the sample space.

  13. An Example • When rolling a single die, what is the probability of… • a) rolling a 2? A = {2}, S = {1,2,3,4,5,6} P(A) = n(A) = 1 n(S) 6

  14. Example #1 (Part 2) • When rolling a single die, what is the probability of… • b) rolling an even number? A = {2,4,6}, S = {1,2,3,4,5,6} P(A) = n(A) = 3 = 1 n(S) 6 2

  15. Example #1 (Part 3) • When rolling a single die, what is the probability of… • c) rolling a number less than 5? A = {1,2,3,4}, S = {1,2,3,4,5,6} P(A) = n(A) = 4 = 2 n(S) 6 3

  16. Example #1 (Part 4) • When rolling a single die, what is the probability of… • d) rolling a number greater than or equal to 5? A = {5,6}, S = {1,2,3,4,5,6} P(A) = n(A) = 2 = 1 n(S) 6 3

  17. The Complement of a Set • The complement of a set A, written A’, consists of all outcomes in the sample space that are not in the set A. • If A is an event in a sample space, the probability of the complementary event, A’, is given by: P(A’) = 1 – P(A)

  18. Example #2 • When selecting a single card from a complete deck (no Jokers), what is the probability you will pick… • a) the 7 of Diamonds? P(A) = n(A) = 1 n(S) 52

  19. Example #2 (Part 2) • When selecting a single card from a complete deck (no Jokers), what is the probability you will pick… • b) a Queen? P(A) = n(A) = 4 = 1 n(S) 52 13

  20. Example #2 (Part 3) • When selecting a single card from a complete deck (no Jokers), what is the probability you will pick… • c) a face card? P(A) = n(A) = 12 = 3 n(S) 52 13

  21. Example #2 (Part 4) • When selecting a single card from a complete deck (no Jokers), what is the probability you will pick… • d) a card that is not a face card? P(A) = n(A) = 40 = 10 n(S) 52 13

  22. Example #2 (Part 5) • Another way of looking at P(not a face card)… • we know: P(face card) = 3 13 • and, we know: P(A’) = 1 - P(A) • So… P(not a face card) = 1 - P(face card) P(not a face card) = 1 - 3 = 10 13 13

  23. Example #2 (Part 6) • When selecting a single card from a complete deck (no Jokers), what is the probability you will pick… • d) a red card? P(A) = n(A) = 26 = 1 n(S) 52 2

  24. Assignment, etc. • Read “Taking a Chance” by Rene Ritson: http://www.infj.ulst.ac.uk/NI-Maths /hypotenuse/volume13/Ritson.html Next class: A look at Venn Diagrams

  25. Probability and Chance

  26. Probability • Probability is a measure of how likely it is for an event to happen. • We name a probability with a number from 0 to 1. • If an event is certain to happen, then the probability of the event is 1. • If an event is certain not to happen, then the probability of the event is 0.

  27. Probability • If it is uncertain whether or not an event will happen, then its probability is some fraction between 0 and 1 (or a fraction converted to a decimal number).

  28. 1. What is the probability that the spinner will stop on part A? B A C D • What is the probability that the spinner will stop on • An even number? • An odd number? 3 1 2 A 3. What fraction names the probability that the spinner will stop in the area marked A? C B

  29. Probability Activity • In your group, open your M&M bag and put the candy on the paper plate. • Put ten brown M&Ms and five yellow M&Ms in the bag. • Ask your group, what is the probability of getting a brown M&M? • Ask your group, what is the probability of getting a yellow M&M?

  30. Examples • Another person in the group will then put in 8 green M&Ms and 2 blue M&Ms. • Ask the group to predict which color you are more likely to pull out, least likely, unlikely, or equally likely to pull out. • The last person in the group will make up his/her own problem with the M&Ms.

  31. Probability Questions • Lawrence is the captain of his track team. The team is deciding on a color and all eight members wrote their choice down on equal size cards. If Lawrence picks one card at random, what is the probability that he will pick blue? blue blue green black yellow blue black red

  32. Donald is rolling a number cube labeled 1 to 6. Which of the following is LEAST LIKELY? • an even number • an odd number • a number greater than 5

  33. CHANCE • Chance is how likely it is that something will happen. To state a chance, we use a percent. ½ Probability 0 1 Equally likely to happen or not to happen Certain to happen Certain not to happen Chance 50 % 0% 100%

  34. Chance • When a meteorologist states that the chance of rain is 50%, the meteorologist is saying that it is equally likely to rain or not to rain. If the chance of rain rises to 80%, it is more likely to rain. If the chance drops to 20%, then it may rain, but it probably will not rain.

  35. 1 2 1. What is the chance of spinning a number greater than 1? 4 3 • What is the chance of spinning a 4? • What is the chance that the spinner will stop on an odd number? 4 1 2 3 5 4. What is the chance of rolling an even number with one toss of on number cube?

  36. DRILL • What is the probability of rolling an odd number on a 6-sided die? • What is the probability of getting a green marble, if there is a bag with 6 blue marbles, 5 green marbles, 3 yellow marbles and 8 orange marbles? • What is the probability of not getting a blue or yellow marble from the same bag?

  37. Tree Diagram • Is a method used for writing out all the possible outcomes for multiple events.

  38. Sample Space • The sample space is the set of all possible outcomes for a given event. • Example: The sample space for rolling a die is {1, 2, 3, 4, 5, 6}

  39. Counting Principle • If two or more events occur in x and y ways to find the total number of combinations (choices) you simply multiply the number of possible outcomes in each group by each other. • Example: If you have 4 shirts, 3 pairs of pants, 2 pairs of shoes and 3 hats, you can make 4(3)(2)(3) different outfits. • Which gives you a total of 72 outfits.

  40. Factorial • Is used when you want to figure out how many ways “n” number of objects can be arranged. • The symbol for factorial is an exclamation point. (n!) • Factorial means to multiply by every number less then “n” down to 1. • Example: 5! = 5(4)(3)(2)(1)

  41. DRILL • What is the probability of rolling a number less than 5 on a 6-sided die? • What is the probability of getting a green marble, if there is a bag with 4 blue marbles, 3 green marbles, 4 yellow marbles and 9 orange marbles? • What is the probability of not getting a green or yellow marble from the same bag?

  42. Classwork Pages 756 – 757 #’s 1 – 22, 25, 26

  43. DRILL • What is the probability of rolling a number less than 5 on a 6-sided die? • What is the theoretical probability of getting a green marble, if there is a bag with 16 blue marbles, 18 green marbles, 14 yellow marbles and 16 orange marbles? • What is the experimental probability of rolling a 3 given: {2, 3, 4, 1, 3, 4, 6, 5, 2, 3, 3, 4, 1, 6}

  44. Algebra IExperimental vs. Theoretical Probability

  45. Theoretical Probability • The theoretical probability of an event is the “actual” probability of something happening. • Number of “correct” outcomes divided by the total number of outcomes.

  46. Experimental Probability • The experimental probability of an event is the probability of an event based on previous outcomes. • Example: If you flipped a coin 10 times and got { T, T, H, T, H, H, H, T, T, T} • The experimental probability of getting tails is 6 out of 10 or 3/5.

  47. Example • {2, 4, 1, 6, 5, 1, 1, 4, 5, 3, 2, 3, 6, 6} • {1, 4, 5, 5, 3, 3, 6, 2, 6, 6, 2, 3, 4, 1} • {2, 3, 2, 2, 5, 6, 1, 2, 3, 4, 3, 2, 2, 6}

  48. Calculator Activity * We are going to simulate rolling a die 50 times using the calculators and then calculate the theoretical probability and experimental probability of the event.

  49. Homework • Write five events and say what the theoretical probability and experimental probability of the events are. • Ex: {1, 4, 3, 5, 5, 2, 3, 1, 6, 2, 2} • Theoretical Prob of rolling a 2 is 1/6 • Experimental Prob of rolling a 2 is 3/11

  50. Stick with Probability

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