Math 1601 Study Guide: Lesson 4

Designed by Engin A. Sungur

Lesson 4:

This fourth lesson covers ...

Objectives
Key Terms
Reading Assignment
Activities
Chapter 4 Study Questions
Chapter 4 Study Notes
Written Assignment

Objectives

After completing this lesson, you should be able to

phrase an inference about a population by making a jump from sample to population and then to give a measure of reliability for the inference.
understand basic concepts of probability
identify random variables and find probabilities of specific numerical outcomes
find and interpret mean and variance of random variables

Reading Assignment

Read chapter 4 in Introduction to the Practice of Statistics.

Activities

The Detailed Learning Objectives

Chapter 4

4.1 Probability Models

Relative frequency and subjective probability
Determination of sample space (S)
Rules of probability
Finding P(A^c), P(A or B), P(A and B) for disjoint and independent events
Independence (verifying A and B independent or not,
finding probabilities given independence)

4.2 Random Variables

discrete and continuous random variables
given a probability distribution finding event probabilities
constructing a probability distribution

4.3 Mean and Variances of Random Variables

given a probability distribution finding mean, variance and standard deviation

4.4 Probability Laws

finding P(A or B), P(A and B) ingeneral
conditional probabilities (finding )
construction of a tree diagram
Bayes's Rule

Key Terms

probability model
sample space
event
complement of an event
disjoint events
independent events
complement rule
addition rule
multiplication rule
random variable
discrete random variable
continuous random variable
density curve
Normal distribution
mean
standard deviation
law of large numbers
tree diagram
Bayes's rule

Chapter 4: Study Questions

What is the difference between sample space and event?
What is the difference between disjoint events and independent events?
What are complement, addition and multiplication rules?
What does a probability distribution of a random variable give us?
What is the difference between discrete and continuous random variables?
What is the density curve?
What is the Normal distribution?
What is the mean and standard deviation of a random variable?
What is conditional probability?
What is the use of Bayes's rule?

Chapter 4: Study Notes

A phenomenon is random if

individual outcome is uncertain
there is a regular distribution of outcomes in a large number of repetitions

Examples Toss a coin, choose a random sample

Probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.
Here is how it is done historically:

The Person who Tossed the Coin	Number of Tosses	Number of Heads	Relative Frequency
Buffon (naturalist) Coin	4040	2048	2048/4040=0.5069
Pearson (statistician)	24000	12012	12012/24000=0.5005
Kerrich (mathematician, in prison)	10000	5067	5067/10000=0.5067

Note that if we go with this definition of probability one should be able to repeat the exreiment under the same condition many times. Do you think that it is possible to do this for all the cases? For example, you need to figure out the probability that you will pass this course. How would you figure this probability? The definition given above is the one that is introduced by the school of probabilist who are relative frequenist. Other approach to the probability is known as personal probability or subjective probability approach. This school believes that a probability is assigned to an event based on subjective judgement, experience, information, and belief.
Whichever appraoch one selects to use there should be a common framework. In section 4.2 we will develop this common framework.

4.2. and 4.5. PROBABILITY MODELS

We start with an experiment and define the following:
Sample Space is the set of all possible outcomes of an experiment, which is represented by S.
Event is any subset of the sample space. In other words it a any combination of possible outcomes of the experiment. Generally, they are represented by capital letters, such as A, B, C, ....

Here are some examples:

Experiment: Toss a coin one time
Sample Space: S={H,T}
An Event: A=getting a head={H}

Experiment: Toss a coin three times
Sample Space: S={HHH,THH,HTH,HHT,TTH,THT,HTT,TTT}
An Event: A="getting two heads"={THH,HTH,HHT}
Another event could be
B="longest run of tails being 2"={TTH,HTT}

Sometimes a "tree" could help us to understand the sample space better.
Experiment: Flip a coin if head occurs flip it for the second time. If tail occurs toss a die.

Sample Space: S={HH,HT,T1,T2,T3,T4,T5,T6}
An Event: A="getting a 3 the die"={T3}
Note that, I did not put A={3}, because 3 is not the element of the sample space.

Experiment: Select a number between 0 and 1
Sample Space: S={all numbers between 0 and 1}
An Event: A="selecting a number less than or equal to the median"={all numbers between or equal to 0 and 0,5}

BASIC PROBABILITY RULES

Let us start with the first two rules. These rules help us to end up with legitimate values for the probabilities:
Rule 1: P(A) is always between 0 and 1.
Rule 2: P(S)=1

Here are the steps for assigning probability to an event:

Define the experiment
List all possible outcomes
Assign probabilities to each outcome
Determine the outcomes in the event, say A
Sum the outcome probabilities that are in the event A.

Example:
Experiment: taking a course on statistics
S={A,B,C,D,F,I,W}
the sample space gives us the possible grades.
Based on the past experience the following probabilities are assigned to the outcomes of this experiment:

OUTCOME	A	B	C	D	F	I	W
PROBABILITY	0.2	0.3	0.2	0.1	0.1	0.05	?

The "?" should be 0.05 by the Rule 2.
Say we want to find the probability that a student will get C or better.
H="getting C or better"={A,B,C}
Therefore,
P(H)=P(A)+P(B)+P(C)=0.2+0.3+0.2=0.7.

Finding a probability of an event gets easier. if the sample space includes finite number of outcomes and these outcomes are eqaually likely to occur. Please note that not all sample spaces has equally likely outcomes.
Suppose that there are k possible outcomes, and outcomes are equally likely, then

Probability of each outcome=1/k
P(A)=(Number of outcomes in A)/k

SUMMARY OF BASIC COMBINED EVENTS

Operation	Explanation	Notation	Venn Diagram
Complement of an event A	The event that A does not occur	A^c
Intersection of A and B	Both A and B occur at the same time	A or B
Union of A, B	Either A or B or both occurs At least one of the events A, B occurs	A and B

OTHER RULES OF PROBABILITY

For the other rules of probability we need to distinguish three cases:

Disjoint Events If the events have no common outcome they are called disjoint events. In other words, the venets, say A and B, can not occur together.
Independent Events If whether A occurs or not does not change the probability that B occurs then A and B are called independent.

Note that being disjoint is a property of events itself, but being independent is a property of the probability of events.

Here is the summary of all the rules

	GENERAL CASE	DISJOINT EVENTS	INDEPENDENT EVENTS
Rule 3. Complement Rule	P(A^C)=1-P(A)	P(A^C)=1-P(A)	P(A^C)=1-P(A)
Rule 4. Addition Rule	P(A or B)= P(A)+P(B)-P(A and B)	P(A or B)= P(A)+P(B)	P(A or B)= P(A)+P(B)-P(A)P(B)
Rule 5. Multiplication Rule	P(A and B)= P(A\|B)P(B)	P(A and B)= 0	P(A and B)= P(A)P(B)

Note that general multiplication rule uses conditional probability. To understand this concept better please work on the activity LET US ROLL A DIE. Here is the formal definition of the conditional probability.

Definition The conditional probability of B given A is:

P(B|A)=P(A and B)/P(A)

Note that for independent events

P(B|A)=P(B)

P(A|B)=P(A)

There are two types of questions that one might face related with the independence

You are given that the events are independent and asked to find the probability that all of these events will occur at the same time. In this case you multiply the individual probabilities.
Based on the information given verify whether two events A and B are independent or not. Here you need to check whether or not P(A and B) equal to P(A)P(B). Or you can also check whether or not P(A|B) equal to P(A).

MORE ON PROBABILITY TREES AND BAYES'S RULE

To be able to understand the importance of rule one should know the difference between the events A|B and B|A, and one should be aware of the fact that the difficulty of obtaining information on probability of occurrence of these two events may not be the same. For example let A="the person has AIDS", B="the HIV test turns out to be positive". Suppose that you are researcher. Which one of the two is easier to find through experiments P(A|B) or P(B|A)?

If you would like to have more information on "How to Solve Problems Related with Probability?" and "Use of Probability Trees and Bayes's Rule", please click on anywhere on this sentence.

Most of the time Bayes's rule is applied to "testing" problem. Here is an example:

In one of Marilyn Savant's columns in parade Magazine the following question was asked.

Suppose we assume that 5% of the people are drug-users. The test is 95% accurate, which we'll say means that if a person is a user, the result is positive 95% of the time; and if s/he isn't, it's negative 95% of the time. A randomly chosen person tests positive. Is the individual highly likely to be a drug-user?

Marilyn's answer was:

Given your condition, once the person has tested positive, you may as well flip a coin to determine whether she of he is a drug-user. The chances are only 50-50.

If you are having a hard time understanding why, please click on any where on this sentence.

If you would like to see another case where Marilyn was right again, please try the activity TO SWITCH OR NOT TO SWITCH.

4.3 and 4.4 RANDOM VARIABLES AND THEIR MEANS AND VARIANCES

RANDOM VARIABLE: is a variable whose value is a numerical outcome of a random phenomenon.

There are two types of random variables

Discrete Random Variables: These are the ones which takes coutable number of possible values
Continuous Random Variables: These can assume any value in one or more intervals.

Notation: X,Y,Z represent the random variable, x,y.z represent possible values of the random variable.

Here are some examples:

X=shoe size; x=5, 5.5, 6, 6.5, ..., 12 (discrete)
X=life length of a battery; x can be any thing from 0 to infinity (continuous)
X=number of trials to obtain the first head; x=1,2,3,... (discrete)
X=amount of time you need to wait to obtain the first head; x can be any thing from 0 to infinity (continuous)

Probability Distribution of a Discrete Random Variable lists all possible values and their probabilities.

Here are some important things that you should be able to do:

Verify whether or not you are dealing with a legitimate probability distribution. Check all the probabilities. They should be between o and 1. Sum of all the probabilities should be 1.
Construct a probability distribution. By using the knowledge that you have gained in section 4.2 and 4.5, list all possible values and associated probabilities.
Find a probability of an event related with the X. Add the probabilities of the values of X that makes up the event.
Find a probability associated with one possible values of X, given all of the remaining probabilities. Remember that sum of the probabilities should be 1.

When finding mean, variance, and standard deviation of a discrete random variable, constructing a table like the one below may help.

Possible values, x_i	Probability, p_i	x_i p_i	(x_i-m_X)	(x_i-m_X)²	(x_i-m_X)² p_i
…	…	…	…	…	…
…	…	…	…	…	…
…	…	…	…	…	…
…	…	…	…	…	…
…	…	…	…	…	…
…	…	…	…	…	…
		Sum of this column=m= mean			Sum of this column=s² variance

Written Assignment

Do the following assignment. The problems listed are from "Introduction to the Practice of Statistics". When you have worked on the problems and are ready to turn in your findings, click the assignment link below. It will take you to a template where you can fill in your answers to the questions. When you are finished entering your answers, click the submit button, you will be given the location of your completed web page. You may check your assignment responses with your browser at any time, and submit a revision at any time before the due date of the assignment. The due date is Wednesday August 4..

SECTION 4.1.Exercise 4.3 (page 293)

SECTION 4.2.Exercises 4.11, 4.15, 4.21, 4.25, 4.31, 4.33 (pages 306-312)

SECTION 4.3.Exercises 4.41, 4.45 (pages 323, 324)

SECTION 4.4.Exercise 4.63 (page 343)

SECTION 4.5.Exercises 4.79, 4.87, 4.89, (page 360-363)

Lesson Submission 4

Assignment #4.

Proceed to lesson 5

Internet Links

Each day you go online, be sure to check out the Random Statistical Quote for the Day