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Chapter 10: Introducing Probability
STAT 1450
Connecting Chapter 10 to our Current Knowledge of Statistics
Probability theory leads us from data collection (Chapter 8 & Chapter 9) to inference.
The rules of probability will allow us to develop models so that we can generalize from our
(properly collected) sample to our population of interest.
10.0 Introducing Probability
The Idea of Probability
▸ We can trust random samples and randomized comparative
experiments because of chance behavior—chance behavior is
unpredictable in the short run but has a regular and predictable pattern
in the long run.
10.1 The Idea of Probability
Technology Tips—Generating Random Numbers
▸ TI–83/84
Apps Prob Sim (press any key) Toss Coins.
Click Toss. Select +1 (this is the Window key) (for your 2nd trial).
Repeat selecting +1 (this is the Window key) 8 more times to obtain your first 10 trials.
Now select +50 (the Trace key).
Click on the Right Arrow key (to reveal the number of Tales).
Click on the Right Arrow key again (to reveal the number of Heads).
Repeat the last 3 steps to obtain the results for trials 61 – 110.
Technology Tips—Generating Random Numbers
Technology Tips—Generating Random Numbers
▸ JMP
Enter 1 into the top row of Column 1.
Right-click on Column 1. Select Formula.
Under the Functions (grouped) menu Select Random Random Binomial.
In the fields provided, ENTER the: sample size (10 for the first time, then 50) and the
probability (.50).
Click Apply. This returns the number of heads for the first 10 flips.
Repeat the process with Column 2. Use a sample size of 50 instead of 10.
Technology Tips—Generating Random Numbers
Predicting Chance Behavior
▸ Example: Let’s observe the behavior of 10 coin flips.
10.1 The Idea of Probability
# of Flips
Expected Proportion of
HeadsActual
# of HeadsActual
# of TailsActual
Proportion of Heads10 .50
Complete the chart for 60 and 110 flips.
60 .50
110 .50
Predicting Chance Behavior
▸ Example: Let’s observe the behavior of 10 coin flips.
10.1 The Idea of Probability
# of Flips
Expected Proportion of
HeadsActual
# of HeadsActual
# of TailsActual
Proportion of Heads10 .50 4 6 .40
Complete the chart for 60 and 110 flips.
60 .50110 .50
Predicting Chance Behavior
▸ Example: Let’s observe the behavior of 10 coin flips.
10.1 The Idea of Probability
# of Flips
Expected Proportion of
HeadsActual
# of HeadsActual
# of TailsActual
Proportion of Heads10 .50 4 6 .40
Complete the chart for 60 and 110 flips.
60 .50 27 33 .55
110 .50
Predicting Chance Behavior
▸ Example: Let’s observe the behavior of 10 coin flips.
10.1 The Idea of Probability
# of Flips
Expected Proportion of
HeadsActual
# of HeadsActual
# of TailsActual
Proportion of Heads10 .50 4 6 .40
Complete the chart for 60 and 110 flips.
60 .50 27 33 .45
110 .50 58 52 .527
Predicting Chance Behavior
▸ What should happen to the long run proportion of heads as we take
more flips?
▸ Probability theory informs us that the sample proportion of heads
should ‘converge’ to the population proportion (50%).
10.1 The Idea of Probability
Key Terminology—Randomness and Probability
▸ We call a phenomenon random if individual outcomes are uncertain
but there is nonetheless a regular distribution of outcomes in
a large number of repetitions.
▸ The probability of any outcome of a random phenomenon is the
proportion of times the outcome would occur in a very long series of
repetitions.
10.2 Randomness and Probability
Randomness and Probability
▸ Technically, physically tossing a coin or rolling a die is predictable. If theoretically, the same force, angles, etc… are applied, the results should remain the same. But, usually persons apply differing amounts to these variables, thus resulting in what appears to be a random process.
▸ Using technology simulations (via graphing calculators or JMP) restore the randomness to the process.
10.2 Randomness and Probability
Randomness and Probability
▸ Technically, physically tossing a coin or rolling a die is predictable. If theoretically, the same force, angles, etc… are applied, the results should remain the same. But, usually persons apply differing amounts to these variables, thus resulting in what appears to be a random process.
▸ Using technology simulations (via graphing calculators or JMP) restore the randomness to the process.
10.2 Randomness and Probability
Poll
▸ Enter your actual proportion of heads after 60 trials _____________
▸ As the number of coin flips increases, the proportion of heads should become closer to 50%.
a) True b) False
10.2 Randomness and Probability
Poll
▸ Enter your actual proportion of heads after 60 trials _____________
▸ As the number of coin flips increases, the proportion of heads should become closer to 50%.
a) True b) False
10.2 Randomness and Probability
Key Terminology—Probability Models
▸ The sample space S of a random phenomenon is the set of all
possible outcomes.
▸ An event is an outcome or a set of outcomes of a random
phenomenon. That is, an event is a subset of the sample space. For
an event A, the probability that A occurs is denoted P(A).
▸ A probability model is a mathematical description of a random
phenomenon consisting of two parts:
a sample space S and a way of assigning probabilities to events.
10.3 Probability Models
Example: Sales Effectiveness ▸ In a sales effectiveness seminar a group of sales representatives tried
two approaches to selling a customer a new automobile: the aggressive approach and the passive approach.
10.3 Probability Models
Result
Sale No Sale Totals
Approach Aggressive 280 260 540
Passive 305 155 460
Totals 585 415 1000
Example: Sales Effectiveness
▸ Sample space of Approach & Result:
{Aggressive-Sale, Aggressive – No Sale, Passive – Sale, Passive – No Sale}
▸ Event:
▸ Probability model:
10.3 Probability Models
Result
P(Result)
Example: Sales Effectiveness
▸ Sample space of Approach & Result:
{Aggressive-Sale, Aggressive – No Sale, Passive – Sale, Passive – No Sale}
▸ Event:
Sale of automobile
▸ Probability model:
10.3 Probability Models
Result
P(Result)
Example: Sales Effectiveness
▸ Sample space of Approach & Result:
{Aggressive-Sale, Aggressive – No Sale, Passive – Sale, Passive – No Sale}
▸ Event:
Sale of automobile
▸ Probability model:
10.3 Probability Models
Result Sale
P(Result) 0.585
Example: Sales Effectiveness
▸ Sample space of Approach & Result:
{Aggressive-Sale, Aggressive – No Sale, Passive – Sale, Passive – No Sale}
▸ Event:
Sale of automobile
▸ Probability model:
10.3 Probability Models
Result Sale No Sale
P(Result) 0.585 0.415
Proportions and Probabilities
▸ Beginning with Chapter 10, we will transition away from what proportion of persons have some characteristic to what is the probability that some event occurs.
10.3 Probability Models
Basic Rules of Probability
▸ The probability of each event assumes a number on the
closed interval [0,1].
▸ All outcomes of the sample space must total to 1.
▸ Addition Rule for disjoint events.
Two events are disjoint if they have no events in common.
In these cases P(A or B)= P(A) + P(B).
▸ Complement Rule
For any event A,
P(A does not occur) = 1 – P(A).
10.4 Probability Rules
Example: Sales Effectiveness
▸ We are familiar with these rules already. From the earlier example
regarding result of sale, P(Aggressive) = 54%.
10.4 Probability Rules
Example: Sales Effectiveness
▸ We are familiar with these rules already. From the earlier example
regarding result of sale, P(Aggressive) = 54%.
Using the complement rule,
P(not Aggressive) = 1- .54 = .46 = P(Passive)
10.4 Probability Rules
Finite and Discrete Probability Models
Discrete probability models have countable outcomes.
These outcomes either assume fixed values or are Natural
numbers {0, 1, 2, …}.
10.5 Finite and Discrete Probability Models
Finite and Discrete Probability Models
▸ Example: Within the next 30 seconds, write down as many Michael
Jackson songs as possible.
10.5 Finite and Discrete Probability Models
Continuous Probability Models
▸ A continuous probability model assigns probabilities as areas under
a density curve.
▸ The area under the curve and above any range of values is the
probability of an outcome in that range.
10.6 Continuous Probability Models
Continuous Probability Models
Recall from Chapter 3:
A density curve is the overall pattern of a distribution.The area under the curve for a given range of values
along the x- axis is the proportion of the population that falls in that range.
A density curve has total area 1 underneath it.
10.6 Continuous Probability Models
Density Curves
▸ Example (from Chp. 3): Despite any rare arctic blasts, January high temperatures in Columbus tend to be uniformly distributed between 35 and 40 degrees.
height =
35 40
b. What proportion of the time is the January hi temperature below 38 degrees? 38
Density Curves
▸ Example (from Chp. 3): Despite any rare arctic blasts, January high temperatures in Columbus tend to be uniformly distributed between 35 and 40 degrees.
height =
35 40
b. What proportion of the time is the January hi temperature below 38 degrees?
Area= (base)(height) = (38-35)*(1/5)
38
Density Curves
▸ Example (from Chp. 3): Despite any rare arctic blasts, January high temperatures in Columbus tend to be uniformly distributed between 35 and 40 degrees.
height =
35 40
b. What proportion of the time is the January hi temperature below 38 degrees?
Area= (base)(height) = (38-35)*(1/5)
38
=3*.20 P(X<38)=.60
Example: Length of a Michael Jackson Song
▸ What is the sample space for the length (in minutes) of a Michael
Jackson song?
Thriller 5:57 Billie Jean 4:54 Smooth Criminal 4:17 Black or
White 3:18
S = {all numbers between 0 and infinity?} (0,
▸ Continuous variables can assume all numerical values over a
particular range.
10.6 Continuous Probability Models
Random Variables
▸ A random variable is a variable whose value is a numerical outcome
of a random phenomenon.
▸ The probability distribution of a random variable X tells us what
values X can take and how to assign probabilities to those values.
▸ Random variables can be discrete or continuous.
Discrete random variables have a finite list of possible outcomes.
Continuous random variables can take on any value in an interval, with probabilities
given as areas under a curve.
10.7 Random Variables
Poll
Many dentists completed anatomy coursework as undergraduates. Two variables are of extreme interest to them– the number of adult teeth a patient has and the size (mm) of their incisors.
Are both of these variables continuous random
variables?
Why or why not?
10.7 Random Variables
Poll
Many dentists completed anatomy coursework as undergraduates. Two variables are of extreme interest to them– the number of adult teeth a patient has and the size (mm) of their incisors.
Are both of these variables continuous random
variables?
Why or why not?
No. the number of adult teeth is discrete. The size of their incisors is a continuous.
10.7 Random Variables
Poll
Many dentists completed anatomy coursework as undergraduates. Two variables are of extreme interest to them– the number of adult teeth a patient has and the size (mm) of their incisors.
Are both of these variables continuous random
variables?
Why or why not?
No. the number of adult teeth is discrete. The size of their incisors is a continuous
1. Number of songs in your iPod?
2. The lengths of each song on your iPod?
10.7 Random Variables
Poll
Many dentists completed anatomy coursework as undergraduates. Two variables are of extreme interest to them– the number of adult teeth a patient has and the size (mm) of their incisors.
Are both of these variables continuous random
variables?
Why or why not?
No. the number of adult teeth is discrete. The size of their incisors is a continuous
1. Number of songs in your iPod? Discrete
2. The lengths of each song on your iPod?
10.7 Random Variables
Poll
Many dentists completed anatomy coursework as undergraduates. Two variables are of extreme interest to them– the number of adult teeth a patient has and the size (mm) of their incisors.
Are both of these variables continuous random
variables?
Why or why not?
No. the number of adult teeth is discrete. The size of their incisors is a continuous
1. Number of songs in your iPod? Discrete
2. The lengths of each song on your iPod? Continuous
10.7 Random Variables
Personal Probability
A personal probability is a number between 0 and 1 that
expresses
someone’s judgment of an event’s likelihood.
Example: I believe there is a 20% chance of precipitation
tomorrow.
This is based upon no relative frequency, no scientific evidence, just intuition.
10.8 Personal Probabilities
Five-Minute Summary
▸ List at least three concepts that had the most impact on your
knowledge of probability.
_____________ ________________
_________________
