Chapter 10: Introducing Probability STAT 1450. Connecting Chapter 10 to our Current Knowledge of...

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.

_____________ ________________

_________________