1. Introduction to Simulations and Experimental Probability
Introduction to Probability

2. 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

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 )
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.
/Gerolamo Cardano
/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)

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)

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)

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)

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)

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) = = 1
n(S)

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)

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)

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) = = 10

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)

24. Assignment, etc. Read “Taking a Chance” by Rene Ritson:
/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?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
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.
Probability 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 I Experimental 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

51. Objectives:
Students will learn new vocabulary terms relating to probability.
Students will determine the probability of different events.
Students will develop a pie chart for their probability data.

52. Vocabulary Probability Certain Impossible Likely Unlikely
Possible Outcomes
Experimental Probability

53. Vocabulary
Probability: chance that an event will happen (study of chance)
Certain: that event will happen
Impossible: that event will never happen
Likely: greater chance that it will happen, than not happen
Unlikely: greater chance that it will not happen, than it will happen
Possible Outcomes: number of outcomes you are able to achieve
Experimental Probability: can be found by conducting repeated trials

54. Probability of an event….
number of favorable outcomes
number of possible outcomes

55. Demonstration….

56. What would be the probability of picking a green marble?5 8

57. If we have a bag of colored sticks?
1. What is the probability in and getting a yellow?
2. Green?
3. Red?
4. Blue?

58. More Questions???
5. How could you change the sticks in the bag so that the chances are just as good at getting one color as it is another?
6. What color has the greatest chance of being picked?
7. Least chance of being picked?

59. Probability Activity…. Have Fun!
4-5 Groups
bag of colored sticks (don’t peak!)
*two yellow
*seven blue
*three red
*five green
tally chart
dry-erase markers (only on the chart!)

60. Directions….
You are going to (as a group) draw out of the bag 15 times, taking turns.
Each time you draw a stick, draw a tally under the color you drew on the chart.
Afterwards, add each color up to get the total. Write these totals under each color in the indicated box.
TOMORROW… we will work on the computer and make a pie graph that represents our data.
GOOD LUCK!

61. What are some other probability tools?
Dice (number cubes)
Colored Candies
Colored Pipe Cleaners
Marbles
Spinner
Coin Toss

62. Resources

63. 9-3 Use a Simulation Warm Up Problem of the Day Lesson Presentation
Pre-Algebra

64. 9-3 Use a Simulation Warm Up 0.12 0.52
Pre-Algebra
9-3
Use a Simulation
Warm Up
1. There are 25 out of 216 sophomores enrolled in a physical-education course. Estimate the probability that a randomly selected sophomore is enrolled in a physical-education course.
2. A spinner was spun 230 times. It landed on red 120 times, green 65 times, and yellow 45 times. Estimate the probability of its landing on red.
0.12
0.52

65. Problem of the Day
If a triangle is worth 7 and a rectangle is worth 8, how much is a hexagon worth? 10

66. Learn to use a simulation to estimate probability.

67. Vocabulary
simulation
random numbers

68. A simulation is a model of a real situation
A simulation is a model of a real situation. In a set of random numbers, each number has the same probability of occurring as every other number, and no pattern can be used to predict the next number. Random numbers can be used to simulate random events in real situations.

69. Understand the Problem
Additional Example 1: Problem Solving Application
A dart player hits the bull’s-eye 25% of the times that he throws a dart. Estimate the probability that he will make at least 2 bull’s-eyes out of his next 5 throws. 1
Understand the Problem
The answer will be the probability that he will make at least 2 bull’s-eyes out of his next 5 throws.
List the important information:
The probability that the player will hit the bull’s-eye is 0.25.

70. Additional Example 1 Continued2
Make a Plan
Use a simulation to model the situation. Use digits grouped in pairs. The numbers 01–25 represent a bull’s-eye, and the numbers 26–00 represent an unsuccessful attempt. Each group of 10 digits represent one trial.

71. Additional Example 1 Continued2
Make a Plan

72. Additional Example 1 Continued
Solve 3
Starting on the third row of the table from the previous slide and using 10 digits for each trial yields the data at right:

73. Additional Example 1 Continued
Out of the 10 trials, 2 trials represented two or more bull’s-eyes. Based on this simulation, the probability of making at least 2 bull’s-eyes out of his next 5 throws is about , or 20%. 2 10
Look Back 4
Hitting the bull’s-eye at a rate of 20% means the player hits about 20 bull’s-eye out of every 100 throws. This ratio is equivalent to 2 out of 10 throws, so he should make at least 2 bull’s-eyes most of the time. The answer is reasonable.

74. Understand the Problem
Try This: Example 1
Tuan wins a toy from the toy grab machine at the arcade 30% of the time. Estimate the probability that he will win a toy 1 time out of the next 3 times he plays. 1
Understand the Problem
The answer will be the probability that he will win 1 of the next 3 times.
List the important information:
The probability that Tuan will win is 30%.

75. Try This: Example 1 Continued2
Make a Plan
Use a simulation to model the situation. Use digits grouped in pairs. The numbers 01–30 represent a win, and the numbers 31–00 represent an unsuccessful attempt. Each group of 6 digits represent one trial.

76. Try This: Example 1 Continued86 58 52 79 19 65 26 49 35 57 94 42 51 33 25 16 63 85 84 18 39 47 32 66 67 89 93 87 83
Solve 3
1 win
Starting on the fourth row of the table from slide 3 and using 6 digits for each trial yields the data at right:
1 win
1 win
1 win

77. Try This: Example 1 Continued
Out of the 10 trials, 4 trials represented one or more wins. Based on this simulation, the probability of winning at least 1 time out of his next 3 games is 40%
Look Back 4
Winning at a rate of 40% means that Tuan wins about 40 times out of every 100 games. This ratio is equivalent to 4 out of 10 games, so he should win at least 4 toys most of the time. The answer is reasonable.

78. Use the table of random numbers to simulate the situation.
Lesson Quiz
Use the table of random numbers to simulate the situation.
38094
76211
43659
29272
76005
93391
19587
47380
33442
40809
27904
95412
69632
48461
25654
55889
42231
39983
13802
24483
52730
15604
80949
46351
10580
59765
76431
38586
62987
40440
93594
30198
64926
17672
68735
35168
19085
35497
30798
21966
Lydia gets a hit 34% of the time she bats. Estimate the probability that she will get at least 4 hits in her next 10 at bats.
Possible answer: 30%

79. Examples
A bag contains 30 marbles 12 red, 8 blue, 4 orange and 6 green. What is the probability that you do not pick a blue.
If you flip two coins 40 times and you got HH 11 times, HT 9 times, TH 12 times and TT 8 times. What is the experimental probability of getting HH?
What is the probability of getting a vowel from the word “MATHEMATICS”

80. Chapter 6 Review Probability TEST TOMMOROW

81. Examples
4) How could you do a simulation for making a shot in basketball if you had a 60% chance of making the shot and you were shooting 5 times?
5) If 12 out of 50 students took carpentry how many would you project take carpentry out of 2000 students?
6) If 22 out of 80 students go on vacation over the summer, how many would you expect to go on vacation if there were 480 students?

82. Examples
7) How many different outfits could you make if you had 5 pairs of pants, 6 shirts and 4 pairs of shoes?
8) If a given teacher has 8 freshman, 14 sophomores, 16 juniors and 22 seniors, what is the probability that the teacher will randomly select one student and it will be a sophomore?
9) What tools could you use to simulate a 40% probability?

83. Examples
10) How many students out of 3,500 would you expect watch over 4 hours of T.V a week if 24 out of 125 students watch over 4 hours of T.V a week?
11) Draw a spinner that might have been used that if it was spun 100 times and it landed on the number one 25 times, the number two 50 times the number three 12 times and the number four 13 times?

84. Examples
ECR