1. Chapter 5
Probability

2. Section
5.1
Probability Rules

3. Objectives Apply the rules of probabilities
Compute and interpret probabilities using the empirical method
Compute and interpret probabilities using the classical method
Use simulation to obtain data based on probabilities
Recognize and interpret subjective probabilities 3

4. Probability is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the long-term proportion with which a certain outcome will occur in situations with short-term uncertainty.
Use the probability applet to simulate flipping a coin 100 times. Plot the proportion of heads against the number of flips. Repeat the simulation. 4

5. Probability deals with experiments that yield random short-term results or outcomes, yet reveal long-term predictability.
The long-term proportion in which a certain outcome is observed is the probability of that outcome. 5

6. The Law of Large Numbers
As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome. 6

7. In probability, an experiment is any process that can be repeated in which the results are uncertain.
The sample space, S, of a probability experiment is the collection of all possible outcomes.
An event is any collection of outcomes from a probability experiment. An event may consist of one outcome or more than one outcome. We will denote events with one outcome, sometimes called simple events, ei. In general, events are denoted using capital letters such as E. 7

8. Consider the probability experiment of having two children.
EXAMPLE Identifying Events and the Sample Space of a Probability Experiment
Consider the probability experiment of having two children.
(a) Identify the outcomes of the probability experiment.
(b) Determine the sample space.
(c) Define the event E = “have one boy”.
e1 = boy, boy, e2 = boy, girl, e3 = girl, boy, e4 = girl, girl
{(boy, boy), (boy, girl), (girl, boy), (girl, girl)}
{(boy, girl), (girl, boy)} 8

9. PRACTICE QUESTION #1
Write your answers on the worksheet.
Consider the probability experiment of flipping 3 coins.
(a) Identify the outcomes of the probability experiment.
(b) Determine the sample space.
(c) Define the event E = “have 2 tails”.

10. Objective 1
Apply Rules of Probabilities 10

11. Rules of probabilities
1. The probability of any event E, P(E), must be greater than or equal to 0 and less than or equal to 1.
That is, 0 ≤ P(E) ≤ 1.
2. The sum of the probabilities of all outcomes must equal 1.
That is, if the sample space S = {e1, e2, …, en}, then P(e1) + P(e2) + … + P(en) = 1 11

12. A probability model lists the possible outcomes of a probability experiment and each outcome’s probability. A probability model must satisfy rules 1 and 2 of the rules of probabilities. 12

13. All probabilities are between 0 and 1, inclusive.
EXAMPLE A Probability Model
In a bag of peanut M&M milk chocolate candies, the colors of the candies can be brown, yellow, red, blue, orange, or green. Suppose that a candy is randomly selected from a bag. The table shows each color and the probability of drawing that color. Verify this is a probability model.
Color
Probability
Brown
0.12
Yellow
0.15
Red
Blue
0.23
Orange
Green
All probabilities are between 0 and 1, inclusive.
Because = 1, rule 2 (the sum of all probabilities must equal 1) is satisfied. 13

14. PRACTICE QUESTION #2
Write your answers on the worksheet.
In a recent survey, middle and high school students were asked about the probability of attending college. Verify this is a probability model.
Grade
Probability
7th
0.83
8th
0.77
9th
0.75
10th
0.68
11th
0.70
12th
0.73

15. If an event is impossible, the probability of the event is 0.
If an event is a certainty, the probability of the event is 1.
An unusual event is an event that has a low probability of occurring. 15

16. Objective 2
Compute and Interpret Probabilities Using the Empirical Method 16

17. Approximating Probabilities Using the Empirical Approach
The probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the experiment.
P(E) ≈ relative frequency of E 17

18. EXAMPLE Building a Probability Model
Outcome
Frequency
Side with no dot
1344
Side with dot
1294
Razorback
767
Trotter
365
Snouter
137
Leaning Jowler 32
Pass the PigsTM is a Milton-Bradley game in which pigs are used as dice. Points are earned based on the way the pig lands. There are six possible outcomes when one pig is tossed. A class of 52 students rolled pigs 3,939 times. The number of times each outcome occurred is recorded in the table at right 18

19. (c) Would it be unusual to throw a “Leaning Jowler”?
EXAMPLE Building a Probability Model
Use the results of the experiment to build a probability model for the way the pig lands.
Estimate the probability that a thrown pig lands on the “side with dot”.
Outcome
Frequency
Side with no dot
1344
Side with dot
1294
Razorback
767
Trotter
365
Snouter
137
Leaning Jowler 32
(c) Would it be unusual to throw a “Leaning Jowler”? 19

20. (a) Outcome Probability Side with no dot Side with dot 0.329 Razorback
0.195
Trotter
0.093
Snouter
0.035
Leaning Jowler
0.008 20

21. (a) Outcome Probability Side with no dot 0.341 Side with dot 0.329
Razorback
0.195
Trotter
0.093
Snouter
0.035
Leaning Jowler
0.008
(b) The probability a throw results in a “side with dot” is In 1000 throws of the pig, we would expect about 329 to land on a “side with dot”. 21

22. (a) Outcome Probability Side with no dot 0.341 Side with dot 0.329
Razorback
0.195
Trotter
0.093
Snouter
0.035
Leaning Jowler
0.008
(c) A “Leaning Jowler” would be unusual. We would expect in 1000 throws of the pig to obtain “Leaning Jowler” about 8 times. 22

23. Outcome Frequency 0 Children 185 1 Child 254 2 Children 359 3 Children
PRACTICE QUESTION #3
Write your answers on the worksheet.
Use the results of the survey to build a probability model for the probability of the number of children a couple will have if they live in a rural community.
Estimate the probability that a couple will have 3 children.
Outcome
Frequency
0 Children
185
1 Child
254
2 Children
359
3 Children
122
4 Children 61
More than 4 Children 12

24. Objective 3
Compute and Interpret Probabilities Using the Classical Method 24

25. The classical method of computing probabilities requires equally likely outcomes.
An experiment is said to have equally likely outcomes when each simple event has the same probability of occurring. 25

26. Computing Probability Using the Classical Method
If an experiment has n equally likely outcomes and if the number of ways that an event E can occur is m, then the probability of E, P(E) is 26

27. Computing Probability Using the Classical Method
So, if S is the sample space of this experiment,
where N(E) is the number of outcomes in E, and N(S) is the number of outcomes in the sample space. 27

28. (a) What is the probability that it is yellow?
EXAMPLE Computing Probabilities Using the Classical Method
Suppose a “fun size” bag of M&Ms contains 9 brown candies, 6 yellow candies, 7 red candies, 4 orange candies, 2 blue candies, and 2 green candies. Suppose that a candy is randomly selected.
(a) What is the probability that it is yellow?
(b) What is the probability that it is blue?
(c) Comment on the likelihood of the candy being yellow versus blue. 28

29. There are a total of 9 + 6 + 7 + 4 + 2 + 2 = 30 candies, so N(S) = 30.
EXAMPLE Computing Probabilities Using the Classical Method
There are a total of = 30 candies, so N(S) = 30.
(b) P(blue) = 2/30 =
(c) Since P(yellow) = 6/30 and P(blue) = 2/30, selecting a yellow is three times as likely as selecting a blue. 29

30. PRACTICE QUESTION #4
Write your answers on the worksheet.
A group of volleyball players consists of three grade-12 students, six grade-11 students, and two grade-10 students. If one player is chosen at random to start a match, what is the probability that she will be from grade 10?

31. Objective 4
Use Simulation to Obtain Data Based on Probabilities 31

32. EXAMPLE Using Simulation
Use the probability applet to simulate throwing a 6-sided die 100 times. Approximate the probability of rolling a 4. How does this compare to the classical probability? Repeat the exercise for 1000 throws of the die. 32

33. Objective 5
Recognize and Interpret Subjective Probabilities 33

34. The subjective probability of an outcome is a probability obtained on the basis of personal judgment.
For example, an economist predicting there is a 20% chance of recession next year would be a subjective probability. 34

35. EXAMPLE Empirical, Classical, or Subjective Probability
In his fall 1998 article in Chance Magazine, (“A Statistician Reads the Sports Pages,” pp ,) Hal Stern investigated the probabilities that a particular horse will win a race. He reports that these probabilities are based on the amount of money bet on each horse. When a probability is given that a particular horse will win a race, is this empirical, classical, or subjective probability?
Subjective because it is based upon people’s feelings about which horse will win the race. The probability is not based on a probability experiment or counting equally likely outcomes. 35

36. PRACTICE QUESTION #4
Write your answers on the worksheet.
Which of the following is true about subjective probability?
It is not calculated mathematically
It can be based on previous experiences
It can be based on a hunch
All of the answers are correct.