Download Chapter 5 - Elementary Probability Theory Historical Background

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
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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
P( E ) 
number of favorable outcomes n( E )

.
total number of outcomes
n( S )
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.
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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
P( E ) 
number of times event E occurred
.
number of times the experiment was performed
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
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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}.
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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”
•
•
•
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
Meaning of “A or B” Union of Sets – “OR”
•
•
•
•
•
•
•
•
•
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
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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.
(The probability of an event is between 0 and 1, inclusive.)
1. 0  P( E )  1
2. P ()  0
(The probability of an impossible event is 0.)
3. P ( S )  1
(The probability of a certain event is 1.)
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.
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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.
P(not E )  P( S )  P( E )
 1  P( E )
So we have P( E )  P  E    1
and
P( E )  1  P( E).
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?
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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 P( A or B)  P( A)  P( B)  P( A and B).
If A and B are mutually exclusive, then P( A or B)  P( A)  P( B).
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?
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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)
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Conditional Probability Formula
The conditional probability of B, given A, and is given by
P( B | A) 
P( A B) P( A and B)

.
P( A)
P( A)
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).
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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
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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
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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.
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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.
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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!
c) (6-3)!
b) 6!
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?
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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
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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
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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
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