chapter 5 - elementary probability theory historical...

1 | P a g e Hannah Province – Mathematics Department Southwest Tennessee Community College

Chapter 5 - Elementary Probability Theory

Historical Background

Much of the early work in probability concerned games and gambling. One of the first to apply

probability to matters other than gambling was Pierre Simon de Laplace, who is often credited with

being the “father” of probability theory. In the twentieth century a coherent mathematical theory of

probability was developed through people such as Chebyshev, Markov, and Kolmogorov.

Probability

The study of probability is concerned with random phenomena. Even though we cannot be certain

whether a given result will occur, we often can obtain a good measure of its likelihood, or probability.

In the study of probability, any observation, or measurement, of a random phenomenon is an

experiment. The possible results of the experiment are called outcomes, and the set of all possible

outcomes is called the sample space.

Usually we are interested in some particular collection of the possible outcomes. Any such subset of the

sample space is called an event.

Example: Tossing a Coin

If a single fair coin is tossed, find the probability that it will land heads up.

Definition: A phenomenon is Random if individual outcomes are uncertain – cannot be determined

beforehand but there is a regular distribution of outcomes in a large number of repetitions of the

experiment.

Probability is the relative frequency of an outcome over a long series of repetitions of the event.

Probability is a numerical measure between 0 and 1 that describes the likelihood that an event will

occur


Probability

• Probability is a numerical measure that indicates the likelihood of an event.

• All probabilities are between 0 and 1, inclusive.

• A probability of 0 means the event is impossible.

• A probability of 1 means the event is certain to occur.

• Events with probabilities near 1 are likely to occur.

• Events can be named with capital letters: A, B, C…

• P(A) means the probability of A occurring.

– P(A) is read “P of A”

– 0 ≤ P(A) ≤ 1

– P(A) = 1, the event is certain to occur

– P(A) = 0, the event is certain not to occur

Theoretical Probability Formula

If all outcomes in a sample space S are equally likely, and E is an event within that sample space, then

the theoretical probability of the event E is given by

Example - Among a sample of 50 dog owners, 23 feed their dogs Mighty Mutt dry dog food.

Estimate the probability that a dog owner selected at random feeds their dogs Mighty Mutt dry food.

a). 23/50 b). 27/50 c). 1/23 d). 23/27

Example: Flipping a Cup

A cup is flipped 100 times. It lands on its side 84 times, on its bottom 6 times, and on its top 10 times.

Find the probability that it will land on its top.

number of favorable outcomes ( )( ) .

total number of outcomes ( )

n EP E

n S


Empirical Probability Formula

If E is an event that may happen when an experiment is performed, then the empirical probability of

event E is given by

Example: Card Hands

There are 2,598,960 possible hands in poker. If there are 36 possible ways to have a straight flush, find

the probability of being dealt a straight flush.

Example: Gender of a Student

A school has 820 male students and 835 female students. If a student from the school is selected at

random, what is the probability that the student would be a female?

Law of Large Numbers

As an experiment is repeated more and more times, the proportion of outcomes favorable to any

particular event will tend to come closer and closer to the theoretical probability of that event.

Example: Toss a coin repeatedly. The relative frequency gets closer and closer to P(head) = 0.50

Relative

Frequency 0.52 0.518 0.495 0.503 0.4996

n = number of

heads 104 259 495 1006 2498

f = number of

flips 200 500 1000 2000 5000

number of times event occurred( ) .

number of times the experiment was performed

EP E


Probability in Genetics

Gregor Mendel, an Austrian monk used the idea of randomness to establish the study of genetics. To

study the flower color of certain pea plants he found that: Pure red crossed with pure white produces

red.

Mendel theorized that red is “dominant” (symbolized by R), while white is recessive (symbolized by r).

The pure red parent carried only genes for red (R), and the pure white parent carried only genes for

white (r).

Every offspring receives one gene from each parent which leads to the tables below. Every second

generation is red because R is dominant.

Example: Probability of Flower Color

Referring to the 2nd to 3rd generation table (previous slide), determine the probability that a third

generation will be

a) red b) white

Base the probability on the sample space of equally likely outcomes: S = {RR, Rr, rR, rr}.


Intersection of Sets – The intersection of sets A and B, written A⋂B, is the set of elements common in

both A and B, or

Union of Sets – The union of sets A and B, written A⋃B, is the set of elements belonging to wither of the

sets, or

Disjoint Sets – Two sets have no elements in common, sets A and B are disjoint if A⋂B = ø.

Meaning of “A and B” - Intersection of Sets – “AND”

•

•

•

Meaning of “A or B” Union of Sets – “OR”

•

•

•

•

•

•

•

•

•

The Venn diagram to the left shows two sets A and

B the intersection of the two sets is the gray

portion, , U is the universe

The Venn diagram to the left shows two sets A and

B the union of the two sets is the gray portion,

, U is the universe


Events Involving “Not” and “Or”

Properties of Probability

Let E be an event from the sample space S. That is, E is a subset of S. Then the following properties

hold.

Example: Rolling a Die

When a single fair die is rolled, find the probability of each event.

a) the number 3 is rolled

b) a number other than 3 is rolled

c) the number 7 is rolled

d) a number less than 7 is rolled

Events Involving “Not”

The table on the next slide shows the correspondences that are the basis for the probability rules

developed in this section. For example, the probability of an event not happening involves the

complement and subtraction.

1. 0 ( ) 1

2. ( ) 0

3. ( ) 1

P E

P

P S

(The probability of an event is between 0 and 1, inclusive.)

(The probability of an impossible event is 0.)

(The probability of a certain event is 1.)


Correspondences

Set Theory Logic Arithmetic

Operation or

Connective (Symbol)

Complement Not Subtraction

Operation or

Connective (Symbol)

Union Or Addition

Operation or

Connective (Symbol)

Intersection And Multiplication

Probability of a Complement

The probability that an event E will not occur is equal to one minus the probability that it will occur.

Example: Complement

When a single card is drawn from a standard 52-card deck, what is the probability that is will not be an

ace?

(not ) ( ) ( )

1 ( )

P E P S P E

P E

So we have

and

( ) 1

( ) 1 ( ).

P E P E

P E P E


Events Involving “Or”

Probability of one event or another should involve the union and addition.

Mutually Exclusive Events

Two events A and B are mutually exclusive events if they have no outcomes in common. (Mutually

exclusive events cannot occur simultaneously.)

Addition Rule of Probability (for A or B)

If A and B are any two events, then

If A and B are mutually exclusive, then

Example: Probability Involving “Or”

When a single card is drawn from a standard 52-card deck, what is the probability that it will be a king or

a diamond?

Example: Probability Involving “Or”

If a single die is rolled, what is the probability of a 2 or odd?

( or ) ( ) ( ) ( and ).P A B P A P B P A B

( or ) ( ) ( ).P A B P A P B


Probability versus Statistics

• Probability is the field of study that makes statements about what will occur when a sample is

drawn from a known population.

• Statistics is the field of study that describes how samples are to be obtained and how inferences

are to be made about unknown populations.

Conditional Probability; Events Involving “And”

Conditional Probability

Sometimes the probability of an event must be computed using the knowledge that some other event

has happened (or is happening, or will happen – the timing is not important). This type of probability is

called conditional probability.

The probability of event B, computed on the assumption that event A has happened, is called the

conditional probability of B, given A, and is denoted P(B | A).

Example: Selecting From a Set of Numbers

From the sample space S = {2, 3, 4, 5, 6, 7, 8, 9}, a single number is to be selected randomly. Given the

events

A: selected number is odd, and

B selected number is a multiple of 3.

find each probability.

a) P(B)

b) P(A and B)

c) P(B | A)


Conditional Probability Formula

The conditional probability of B, given A, and is given by

Example: Probability in a Family

Given a family with two children, find the probability that both are boys, given that at least one is a boy.

Mutually Exclusive Events

• Two events are mutually exclusive if they cannot occur at the same time.

• Mutually Exclusive = Disjoint

• If A and B are mutually exclusive, then

P(A and B) = 0

Addition Rules

• If A and B are mutually exclusive, then

P(A or B) = P(A) + P(B).

• If A and B are not mutually exclusive, then

P(A or B) = P(A) + P(B) – P(A and B).

( ) ( and )( | ) .

( ) ( )

P A B P A BP B A

P A P A


Independent Events

Two events A and B are called independent events if knowledge about the occurrence of one of them

has no effect on the probability of the other one, that is, if

P(B | A) = P(B), or equivalently

P(A | B) = P(A).

P(A | B) denotes the probability that event A will occur given that event B has occurred. This is called

conditional probability. Read “Probability of A given B”.

• Two event s A and B are called dependent events if the occurrence of event B has changed the

probability that event A will occur, that is

Example: Checking for Independence

A single card is to be drawn from a standard 52-card deck. Given the events

A: the selected card is an ace

B: the selected card is red

a) Find P(B).

b) Find P(B | A).

c) Determine whether events A and B are independent.

Multiplication Rules


Example: Multiplication, Independent Events

Suppose you are going to throw two fair die. What is the probability of getting a 5 on each die?

Example: Selecting From an Jar of Balls, Dependent Events

Jeff draws balls from the jar below. He draws two balls without replacement. Find the probability that

he draws a red ball and then a blue ball, in that order.

4 red

3 blue

2 yellow


Example: Selecting From an Jar of Balls, Independent Example

Jeff draws balls from the jar below. He draws two balls, this time with replacement. Find the probability

that he gets a red and then a blue ball, in that order.

4 red

3 blue

2 yellow

Critical Thinking

• Pay attention to translating events described by common English phrases into events described

using and, or, complement, or given.

• Rules and definitions of probabilities have extensive applications in everyday lives.


Multiplication Rule for Counting

This rule extends to outcomes involving three, four, or more series of events.

Example - A coin is tossed and a six-sided die is rolled. How many outcomes are possible?

a). 8 b). 10 c). 12 d). 18

Tree Diagrams

• Displays the outcomes of an experiment consisting of a sequence of activities.

– The total number of branches equals the total number of outcomes.

– Each unique outcome is represented by following a branch from start to finish.

Tree Diagrams with Probability

• We can also label each branch of the tree with its respective probability.

• To obtain the probability of the events, we can multiply the probabilities as we work down a

particular branch.


Example - Place five balls in an urn: three red and two blue. Select a ball, note the color, and, without

replacing the first ball, select a second ball.

The Factorial

For any counting number n, the quantity n factorial is given by

Example – Evaluating Expressions Containing Factorials

Evaluate each expression.

a) 3! b) 6!

c) (6-3)! d) 6!-3!

Arrangements of n Distinct objects n!

Te total number of different ways to arrange n distinct objects is n!

Example – Arranging Essays

Erika Berg has seven essays to include in her English 1A folder. In how many different orders can she

arrange them?


Permutations - arrangements are often called permutations, the number of permutations of n distinct

things taken r at a time is denoted nPr Since the number of objects being arranged cannot exceed the

total number available, we assume for our purposes here that r ≤ n. Applying the fundamental counting

principle to arrangements of this type gives:

nPr =

Permutations are to evaluate the number of arrangements of n things taken r at a time, where

repetitions are not allowed, and the order of the items is important.

Example - For a group of seven people, how many ways can four of them be seated in four chairs?

a). 35 b). 3 c). 28 d). 840

Alternative Notations are P(n,r) and

For Example 4P2 means “the number of permutations of 4 distinct things taken 2 at a time”.

Using a graphing calculator we can perform this calculation directly as follows for a TI-83:

For 10P6 enter in 10 then hit MATH – Scroll over to PRB and scroll down to 2 (nPr) hit enter

then enter in 6 then hit enter.

10P6 = 151200


Example – Using the Factorial Formula for Permutations

Evaluate each permutation

a) 4P2

b) 8P5

Note that 5P5 is equal to 5! It is true for all whole numbers n that nPn = n!

Combinations - the number of combinations of n things taken r at a time (that is the number of size r

subsets, given a set of size n) is written nCr Since there are n things available and we are choosing r of

them, we can read nCr as “n choose r”. The formula for evaluating numbers of combinations

nCr

Permutations are to evaluate the number of arrangements of n things taken r at a time, where

repetitions are not allowed, and the order of the items is important.

Combinations are the number of combinations of n things taken r at a time (that is the number of size r

subsets, given a set of size n), where repetitions are not allowed, and the order is not important.

Factorial Formula for Combinations

The number of combinations, or subsets, of n, distinct things taken r at a time, where r ≤ n, can be

calculated as

nCr

Alternative Notations are C(n,r) and and


Using a graphing calculator we can perform this calculation directly as follows for a TI-83:

For 14C6 enter in 14 then hit MATH – Scroll over to PRB and scroll down to 2 (nCr) hit enter

then enter in 6 then hit enter.

14C6 = 3003

Example –Evaluate each combination

a) 9C7

b) 24C18

Example - Among eleven people, how many ways can eight of them be chosen to be seated?

a). 6,652,800 b). 165

c). 3 d). 88

chapter 5 - elementary probability theory historical...

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