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Chapter 6

Probability, Randomness, and

Uncertainty

Probability, Randomness, and Uncertainty

Sections 6.1-6.4 Classical Probability

Objectives:

• To learn the basic vocabulary used in counting techniques.

• To solve problems using classical probability.



Section 6.1 Important Definitions

Probability:

• In every day usage the word probably describes an event or circumstance the speaker believes will occur. At the same time the speaker reserves the possibility that it may not occur.

• Probability is used to quantify uncertainty.

• Games of chance, such as tossing a coin, provide a way to demonstrate some of the fundamental laws (rules) of probability.

• Statistically speaking, the playing of the game is called an experiment .




Random Experiment:

• A random experiment is defined as any activity or phenomenon that meets the following conditions: i.

There is a distinct outcome for each trial of the experiment.

ii.

The outcome of the experiment is uncertain.

iii.

The set of all distinct outcomes of the experiment can be specified and is called the sample space .




Events:

• A simple event is any member of the sample space.

If you were to toss a single coin there are only 2 simple events

{Head, Tail} in the experiment.

• An event is a set of simple events.




Sample Space:

• The sample space of the experiment contains every possible outcome that could occur in the experiment.

• For a coin tossing experiment the sample space would be

S = {Head, Tail}.

• The sample space may be called the outcome set.

• The result of a random experiment must be a simple event.




Example:

Experiment 1: Toss a coin three times and observe the number of heads.

Have we met the three conditions of a random experiment?

i.

There will only be one outcome since it is not possible to have exactly one and exactly two heads on the same trial.

ii.

The outcome will be unknown before tossing the three coins.

iii.

The sample space can be specified and is composed of eight simple events,

S={TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}

This experiment meets the conditions of a random experiment.

An event could be obtaining more than one head, which involves the following set of simple events, {THH, HTH, HHT, HHH}.




Example:

Experiment 2: Roll a die and observe the number of dots on the uppermost surface.


Solution: i.

There will only be one outcome.

ii.

The value of the outcome is not known.

iii.

The sample space can be specified and is composed of simple events,

S={1, 2, 3, 4, 5, 6}.


An event could be rolling an even number, which would be given by the set of simple events, {2, 4, 6}.




Example:

Experiment 3: Draw a card from a well shuffled deck consisting of 13 hearts, 13 clubs, 13 spades, and 13 diamonds and observe the suit of the card.


Solution: i.

There will only be one outcome.

ii.

The suit is unknown since the card will be drawn at random.

iii.

The sample space consists of the set of outcomes,

S = {heart, club, spade, diamond}.


An event could be drawing either a spade or a club, which would be given by the set of simple events, {spade, club}.


Relative Frequency:

• Someone who wanted the probability of getting a head on the toss of a coin could toss a coin a large number of times and observe the number of times that a head appeared.

• The probability could be computed as the number of times a head was observed divided by the number of times the coin was flipped. This is the relative frequency interpretation of probability.



Section 6.2 Interpreting Probability: Relative Frequency


• If an experiment is performed n times, under identical conditions, and the event A happens k times, the relative frequency of A is

Relative Frequency of

A

 k n

.

• If the relative frequency stabilizes as n increases, then the relative frequency is said to be the probability of A .





This is a graph of the first 42 flips of the coin. The relative frequency after 42 flips is 36%. This means in 42 flips, the coin landed on heads 36% of the time.

More coin tosses are needed to stabilize the percentage.





Here the flips start at 200 flips. By flip 296, there are 141 heads and 155 tails which equate to a .4764 probability (or relative frequency) of heads.

Although this is slightly less than expected, such a percentage is reasonable considering the randomness of the coin toss.





0

It is clear in this chart that from 1350 to 1450 flips the percentage converges on some point close to .5 and remains very stable.

At flip number 1450, there are 718 heads and 732 tails which equate to a .4952 probability (or relative frequency) of heads.




Statistical Regularity:

Will the observed relative frequency ever reach .5?

• No mathematical or physical law requires the observed relative frequency to ever reach some predetermined level. But if the probability of observing a head is really .5, the observed relative frequency should closely approach the value after a large number of flips.

Relative frequency of the event “getting a head”: A n

718

1450



.4952

• The relative frequency of a head seems to converge to the expected relative frequency. We call this statistical regularity .



Section 6.3 Interpreting Probability: Subjective Approach

Subjective Approach:

• An example of subjective probability is a weatherman predicting the weather.

• The local weather man on channel 2 predicts there is an 80% chance of rain, but the weatherman on channel 4 predicts there is a

66% chance of rain, while lastly on channel 5 the weatherman predicts there is a 33% chance of rain.

Who more accurate?

• It is actually possible that all three weatherman are accurate.

How is this possible?





• If 80% of the days channel 2’s weatherman predicted an 80% chance of rain it actually rained then channel two has very accurate weatherman.

• If 66% of the time it rained when the weatherman at channel 4 predicted a 66% chance of rain, he is as accurate as channel 2.

• Lastly if channel 5’s weatherman predicted a 33% chance of rain and it indeed rained 33% of the time he made that prediction, he is as accurate as channels 2, and 4.





• The subjective viewpoint regards the probability of an event as a measure of the degree of belief that an event has occurred.

• Someone’s degree of belief in some event will depend on his or her life experiences.

• The subjective approach must allow for differences in the degree of belief among reasonable people.

• Subjective probability is criticized since it is not a universally accepted criterion. Two reasonable people might examine the same data and reach different conclusions about their degree of belief about some proposition.



Section 6.4 Interpreting Probability: Classical Approach

Classical Approach:

• Probability can be measured as a simple proportion: the number of simple events that compose the event divided by the number of simple events in the sample space. Namely, the

A

P (A)



number of simple events in

A

total number of simple events in the sample space



Section 6.4 Interpreting Probability: Classical Approach

Example:

In experiment 1 we tossed a coin three times and the number of heads was observed. The sample space consisted of 8 simple events {TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}. Let A b e the event of getting at least one head.

What is the probability of A?

Solution:

Since the event A consists of 7 simple events,

{TTH,THT, THH, HTT, HTH, HHT, HHH}, and there are 8 equally likely simple events in the sample space,

P (A)



7

8

.


Sections 6.5-6.11 Basic Probability Rules

Objectives:

• To calculate the conditional probability of a situation.

• To correctly apply the appropriate probability rule to a given situation.



Section 6.5 What is Probability?

What is Probability?:

• There are several conflicting ideas which seek to define the interpretation of probability. In fact, there is a substantial conflict of ideas between the notions of statistical regularity and degree of belief.

• The theory of probability does not require the interpretation of probabilities, just as in geometry the interpretation of points, lines, and planes are irrelevant.

• Fortunately probability must obey certain laws no matter how probability is defined.


Probability Laws:

Probability Law Number 1

A probability of zero means the event cannot happen.


A probability of one means the event must happen.


All probabilities must be in between zero and one inclusively.


The sum of the probabilities of all simple events equals one.



Section 6.7

What’s the Connection Between Probability and

Statistics?

Probability and Statistics:

• Most of the time when statisticians are working with samples, they try to deduce from the samples the population parameters.

• The process of making judgments about population parameters is called statistical inference .

• Because samples are random, there is no guarantee that the sample accurately represents the population.

• If a sample is not representative, then using the sample mean as a guess (inference) of the population mean would not be very wise.

• Probability is used to assess the quality of our inference.

• All statistical conclusions must be endowed with a degree of uncertainty.

• Because probability is used to assess the reliability of sample inferences, it is the foundation of all inferential statistics.


Probability and Business:

• The probability concept also has many applications in business.

When a manager wonders if dropping a bid price by 5% will increase the probability of winning the bid, she is thinking about chance.

• Probability is also used as a criterion in designing and evaluating product reliability, evaluating insurance, inventory management, project management and in the study of queuing theory (a probability analysis of waiting lines).

• A special kind of statistician called an actuary has emerged to assist in the development of insurance models which quantify uncertainty and aid in business decisions.



Section 6.9 Other Probability Rules

Compound Event:

• A compound event is an event that is defined by combining two or more events.




Compound Events:

• Suppose the marketing director for Sports Illustrated believed that anyone who possessed an income greater than $50,000 and/or subscribed to more than one other sports magazine could potentially be a good prospect for a direct mail marketing campaign.

• Let the events

A = {annual income is greater than $50,000} and

B = {subscribes to more than one other sports magazine}.

• There are several different types of compound events. To illustrate the concepts consider the two events A and B.




Union:

• The union of the events A and B is the set of all outcomes that

A



B



• Notice that the union includes all the points in both A and/or B.




Intersection:

• The intersection of the events A and B is the set of all outcomes that are included in both A and B.



• The symbol for intersection is



.

• Notice that the intersection includes only points in both A and B.

This is the set the marketing director is interested in.




Complement:

• The complement of an event A is the set of all outcomes in the sample space that are not in A.

A

The compliment of

A is the shaded region.

• The complement of a set will be written as A c .

• Notice the complement of A includes all points that are not in A.

• For the event A = {annual income is greater than $50,000}, the complement of A would be

A c = {annual income is less than or equal to $50,000}.




Mutually Exclusive:

• Two points are mutually exclusive if they have no points in common.

A B

• Mutual exclusivity is also called disjointedness. The figure above represents two disjoint events.

• The set relationships of complement, union, and intersection give rise to a number of probability laws.




Example:


The probability of is given by

P

 

P (A)

Consider the event A = {annual income is greater than $50,000}.

Suppose the probability of A was .08.

What is the probability of observing someone whose income is less than or equal to $50,000?

Solution:

P (annual income is less than or equal to $50,000)



P

 

P

 



.92.




Probability Law Number 6:

Union of Mutually Exclusive Events

If the events A and B are mutually exclusive, then

P (A



B)



P (A)



P (B).

A B





The Addition Rule

For any two events A and B,

P (A



B)



P (A)



P (B)



P (A



B)



Section 6.10 Conditional Probability

Conditional Probability:

• The probability that one event will occur, given that some event has occurred or is certain to occur, is a conditional probability .

Example:

• Consider the question of whether cigarette smoking harms those who are indirectly exposed to smoke.

• Suppose that 3 percent of women who do not smoke die of cancer. However, if a non-smoking woman is married to a smoker, the probability of dying of cancer is .08. This is a conditional probability, because the sample space is being limited by some condition – in this case limited to wives of smoking husbands.



Section 6.10 Conditional Probability


• The conditional probability of A, given that B has occurred is

P (A | B)



P (A



B)

.

P (B)

• The events A and B can be reversed in the preceding rule to

• P (A | B)

The vertical bar within a probability statement will always mean given .



Section 6.11 Independence

Independence:

• Two events, A and B, are independent if and only if

P (A | B)



P (A) or P



P (B)

• An extremely important concept in statistical analysis is independence.

• Independence describes a special kind of relationship between two events.

• Two events are said to be independent if knowledge of one event does not provide information about the other event’s occurrence.

• If two events are not independent, they are dependent .



Section 6.11 Independence


Multiplication Rule for Independent Events

If two events, A and B, are independent, then

P (A



B)



P (A)



P (B).

If n events, A

1

,A

2

,…,A n

, are independent, then

P (A

1



A

2

 

A ) n



P (A )

1



P

2

 

P (A ).

n


Section 6.12a Basic Counting Rules

Objectives:

• To learn the five basic counting rules.

• To differentiate between permutation and combination.


Section 6.12 Counting

Fundamental Rule for Counting:

• E

1 is an event with n

1 possible outcomes and E

2 is an event with n

2 possible outcomes. The number of ways the events can occur in sequence is n

1

 n

2

. This principle can be applied for any number of events occurring in sequence.



Example:

A local office supply store offers ballpoint pens from three different manufacturers. Each manufacturer’s pens come in either red, blue, black, or green and either fine or medium tip is available for each color. How many different pens does the store carry?

Solution:

Applying the fundamental rule for counting:

3



4



2



24





number of manufacturers





 color of ink

  types of tips

  different pens




Section 6.12 Counting n Factorial:

• The product factorial .

is a special type of product called a

• Suppose n is a positive whole number. Then, n !

   n



• Note: 0! = 1 by definition.

• n ! is read as “ n factorial”.

• Using this notation,

     



Permutation:

• A permutation is a specific order or arrangement of objects in a set. There are n ! permutations of n unique objects.

•

The number of permutations of n unique objects taken k at a time is:

P k n  n !

 

!

.

• You may see also see this denoted as n

P k or

 

.

• If given n objects, with n

1 alike, n

2 alike, …, n k alike, then the number of distinguishable permutations of all n objects is

 n !

!

! !... !

1 2 3 k

 .



Combination:

• A combination is a collection or grouping of objects where the order is not important.

•

The number of combinations of n unique objects taken k at a time is:

C k n  n !

  .

• It is important to remember that permutations are used when order is important and combinations are used when order is not important.

math courseware specialists

Chapter 6

Probability, Randomness, and

Uncertainty

Relative Frequency of

number of simple events in

total number of simple events in the sample space

The probability of is given by

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Chapter 6

Probability, Randomness, and

Uncertainty

Relative Frequency of

number of simple events in

total number of simple events in the sample space

The probability of is given by

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