Download lecture 20

Eigenvalues!
November 10, 2013
Chapter 10 - Probability
◮
We will begin by looking at mathematical formulations associated
with different outcomes possible in an “experiment”.
Chapter 10 - Probability
◮
We will begin by looking at mathematical formulations associated
with different outcomes possible in an “experiment”.
◮
The Sample Space for an experiment is the list of all possible
outcomes.
Chapter 10 - Probability
◮
We will begin by looking at mathematical formulations associated
with different outcomes possible in an “experiment”.
◮
The Sample Space for an experiment is the list of all possible
outcomes.
Each individual outcome in the sample space is called an
elementary event.
◮
Chapter 10 - Probability
◮
We will begin by looking at mathematical formulations associated
with different outcomes possible in an “experiment”.
◮
The Sample Space for an experiment is the list of all possible
outcomes.
Each individual outcome in the sample space is called an
elementary event.
Given a sample space S, a subset E of the sample space (denoted
E ⊂ S) is called an event.
◮
◮
Chapter 10 - Probability
◮
We will begin by looking at mathematical formulations associated
with different outcomes possible in an “experiment”.
◮
The Sample Space for an experiment is the list of all possible
outcomes.
Each individual outcome in the sample space is called an
elementary event.
Given a sample space S, a subset E of the sample space (denoted
E ⊂ S) is called an event. An even can contain none, some, or all
of the elementary events of S.
◮
◮
Chapter 10 - Probability
◮
We will begin by looking at mathematical formulations associated
with different outcomes possible in an “experiment”.
◮
The Sample Space for an experiment is the list of all possible
outcomes.
Each individual outcome in the sample space is called an
elementary event.
Given a sample space S, a subset E of the sample space (denoted
E ⊂ S) is called an event. An even can contain none, some, or all
of the elementary events of S.
Example Suppose two die are rolled. What elemetary events are in
the event “the die sum to 7”?
◮
◮
◮
Chapter 10 - Probability
◮
We will begin by looking at mathematical formulations associated
with different outcomes possible in an “experiment”.
◮
The Sample Space for an experiment is the list of all possible
outcomes.
Each individual outcome in the sample space is called an
elementary event.
Given a sample space S, a subset E of the sample space (denoted
E ⊂ S) is called an event. An even can contain none, some, or all
of the elementary events of S.
Example Suppose two die are rolled. What elemetary events are in
the event “the die sum to 7”?
Solution: E = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}.
◮
◮
◮
Multiplication Principle
What would the sample space for rolling two die be?
Multiplication Principle
What would the sample space for rolling two die be?Well, first of all, it
would be rather large....
Multiplication Principle
What would the sample space for rolling two die be?Well, first of all, it
would be rather large.... For each possible value the first die could take
on,
Multiplication Principle
What would the sample space for rolling two die be?Well, first of all, it
would be rather large.... For each possible value the first die could take
on, there would be six possible values the second die could take.
Multiplication Principle
What would the sample space for rolling two die be?Well, first of all, it
would be rather large.... For each possible value the first die could take
on, there would be six possible values the second die could take. So the
total number of outcomes would be 6 × 6 = 36.
Multiplication Principle
What would the sample space for rolling two die be?Well, first of all, it
would be rather large.... For each possible value the first die could take
on, there would be six possible values the second die could take. So the
total number of outcomes would be 6 × 6 = 36.
Multiplication Principle
If action 1 can be performed in m ways and action 2 can be perfomed in
n ways, then we can perform action 1 followed by action 2 in m × n ways!
Multiplication Principle
What would the sample space for rolling two die be?Well, first of all, it
would be rather large.... For each possible value the first die could take
on, there would be six possible values the second die could take. So the
total number of outcomes would be 6 × 6 = 36.
Multiplication Principle
If action 1 can be performed in m ways and action 2 can be perfomed in
n ways, then we can perform action 1 followed by action 2 in m × n ways!
We can extend this to as many successive actions as needed...
Multiplication Principle
What would the sample space for rolling two die be?Well, first of all, it
would be rather large.... For each possible value the first die could take
on, there would be six possible values the second die could take. So the
total number of outcomes would be 6 × 6 = 36.
Multiplication Principle
If action 1 can be performed in m ways and action 2 can be perfomed in
n ways, then we can perform action 1 followed by action 2 in m × n ways!
We can extend this to as many successive actions as needed...
Example: Mendel’s Pea Plants
◮
Gregor Mendel conducted experiments to determine the basic
principle of heredity of traits from one generation to the next.
Multiplication Principle
What would the sample space for rolling two die be?Well, first of all, it
would be rather large.... For each possible value the first die could take
on, there would be six possible values the second die could take. So the
total number of outcomes would be 6 × 6 = 36.
Multiplication Principle
If action 1 can be performed in m ways and action 2 can be perfomed in
n ways, then we can perform action 1 followed by action 2 in m × n ways!
We can extend this to as many successive actions as needed...
Example: Mendel’s Pea Plants
◮
◮
Gregor Mendel conducted experiments to determine the basic
principle of heredity of traits from one generation to the next.
He tracked flower color in pea plants - purple or white
Multiplication Principle
What would the sample space for rolling two die be?Well, first of all, it
would be rather large.... For each possible value the first die could take
on, there would be six possible values the second die could take. So the
total number of outcomes would be 6 × 6 = 36.
Multiplication Principle
If action 1 can be performed in m ways and action 2 can be perfomed in
n ways, then we can perform action 1 followed by action 2 in m × n ways!
We can extend this to as many successive actions as needed...
Example: Mendel’s Pea Plants
◮
◮
◮
Gregor Mendel conducted experiments to determine the basic
principle of heredity of traits from one generation to the next.
He tracked flower color in pea plants - purple or white
Suppose Mendel had three pea plant seeds each of which might
grow into a plant with either purple or white flowers.
Multiplication Principle
What would the sample space for rolling two die be?Well, first of all, it
would be rather large.... For each possible value the first die could take
on, there would be six possible values the second die could take. So the
total number of outcomes would be 6 × 6 = 36.
Multiplication Principle
If action 1 can be performed in m ways and action 2 can be perfomed in
n ways, then we can perform action 1 followed by action 2 in m × n ways!
We can extend this to as many successive actions as needed...
Example: Mendel’s Pea Plants
◮
◮
Gregor Mendel conducted experiments to determine the basic
principle of heredity of traits from one generation to the next.
He tracked flower color in pea plants - purple or white
◮
Suppose Mendel had three pea plant seeds each of which might
grow into a plant with either purple or white flowers.
◮
If all three seeds were planted, what is the sample space of all
possible outcomes?
Multiplication Principle
What would the sample space for rolling two die be?Well, first of all, it
would be rather large.... For each possible value the first die could take
on, there would be six possible values the second die could take. So the
total number of outcomes would be 6 × 6 = 36.
Multiplication Principle
If action 1 can be performed in m ways and action 2 can be perfomed in
n ways, then we can perform action 1 followed by action 2 in m × n ways!
We can extend this to as many successive actions as needed...
Example: Mendel’s Pea Plants
◮
◮
Gregor Mendel conducted experiments to determine the basic
principle of heredity of traits from one generation to the next.
He tracked flower color in pea plants - purple or white
◮
Suppose Mendel had three pea plant seeds each of which might
grow into a plant with either purple or white flowers.
◮
If all three seeds were planted, what is the sample space of all
possible outcomes?
Pea Plants Experiment
◮
For each plant there are two possiblities: P (purple) and W (white).
Pea Plants Experiment
◮
◮
For each plant there are two possiblities: P (purple) and W (white).
We can form a tree to enumerate all the possibilities!
Pea Plants Experiment
◮
◮
◮
For each plant there are two possiblities: P (purple) and W (white).
We can form a tree to enumerate all the possibilities!
The total number of all possible outcomes is 2 × 2 × 2 by the
multiplication principle.
Pea Plants Experiment
◮
◮
For each plant there are two possiblities: P (purple) and W (white).
We can form a tree to enumerate all the possibilities!
◮
The total number of all possible outcomes is 2 × 2 × 2 by the
multiplication principle.
◮
The event “exactly two seeds produce plants with purple flowers”
consists of {PPW , PWP, WPP}.
What is the probability of this event?
◮
Pea Plants Experiment
◮
◮
For each plant there are two possiblities: P (purple) and W (white).
We can form a tree to enumerate all the possibilities!
◮
The total number of all possible outcomes is 2 × 2 × 2 by the
multiplication principle.
◮
The event “exactly two seeds produce plants with purple flowers”
consists of {PPW , PWP, WPP}.
What is the probability of this event?
◮
◮
We simply find the ratio of the total number of ways our event can
occur
Pea Plants Experiment
◮
◮
For each plant there are two possiblities: P (purple) and W (white).
We can form a tree to enumerate all the possibilities!
◮
The total number of all possible outcomes is 2 × 2 × 2 by the
multiplication principle.
◮
The event “exactly two seeds produce plants with purple flowers”
consists of {PPW , PWP, WPP}.
What is the probability of this event?
◮
◮
We simply find the ratio of the total number of ways our event can
occur to the total number of possible outcomes in the sample space.
Pea Plants Experiment
◮
◮
For each plant there are two possiblities: P (purple) and W (white).
We can form a tree to enumerate all the possibilities!
◮
The total number of all possible outcomes is 2 × 2 × 2 by the
multiplication principle.
◮
The event “exactly two seeds produce plants with purple flowers”
consists of {PPW , PWP, WPP}.
What is the probability of this event?
◮
◮
We simply find the ratio of the total number of ways our event can
occur to the total number of possible outcomes in the sample space.
So, 83 is the probability that exactly two seeds produce plants with
purple flowers.
Probability of an Event
◮
A sample space is called uniform
Probability of an Event
◮
A sample space is called uniform or equiprobable if each individual
possible outcome in the sample space has the same probability of
occurring as any other outcome.
Probability of an Event
◮
◮
A sample space is called uniform or equiprobable if each individual
possible outcome in the sample space has the same probability of
occurring as any other outcome.
If we have an event E (a subset of the sample space) in a uniform
sample space,
Probability of an Event
◮
◮
A sample space is called uniform or equiprobable if each individual
possible outcome in the sample space has the same probability of
occurring as any other outcome.
If we have an event E (a subset of the sample space) in a uniform
sample space, then the probability of the event occurring is
# of elements in E
total number of possible outcomes in the sample space
.
Probability of an Event
◮
◮
A sample space is called uniform or equiprobable if each individual
possible outcome in the sample space has the same probability of
occurring as any other outcome.
If we have an event E (a subset of the sample space) in a uniform
sample space, then the probability of the event occurring is
# of elements in E
total number of possible outcomes in the sample space
.
Example 2: If three dice are thrown, find the probability that the sum of
the three dice is six.
Probability of an Event
◮
◮
A sample space is called uniform or equiprobable if each individual
possible outcome in the sample space has the same probability of
occurring as any other outcome.
If we have an event E (a subset of the sample space) in a uniform
sample space, then the probability of the event occurring is
# of elements in E
total number of possible outcomes in the sample space
.
Example 2: If three dice are thrown, find the probability that the sum of
the three dice is six.
◮
First we find the size of the sample space:
Probability of an Event
◮
◮
A sample space is called uniform or equiprobable if each individual
possible outcome in the sample space has the same probability of
occurring as any other outcome.
If we have an event E (a subset of the sample space) in a uniform
sample space, then the probability of the event occurring is
# of elements in E
total number of possible outcomes in the sample space
.
Example 2: If three dice are thrown, find the probability that the sum of
the three dice is six.
◮
First we find the size of the sample space: 6 × 6 × 6 = 216 by the
multiplication principle
Probability of an Event
◮
◮
A sample space is called uniform or equiprobable if each individual
possible outcome in the sample space has the same probability of
occurring as any other outcome.
If we have an event E (a subset of the sample space) in a uniform
sample space, then the probability of the event occurring is
# of elements in E
total number of possible outcomes in the sample space
.
Example 2: If three dice are thrown, find the probability that the sum of
the three dice is six.
◮
◮
First we find the size of the sample space: 6 × 6 × 6 = 216 by the
multiplication principle
Next we find all outcomes in our event:
Probability of an Event
◮
◮
A sample space is called uniform or equiprobable if each individual
possible outcome in the sample space has the same probability of
occurring as any other outcome.
If we have an event E (a subset of the sample space) in a uniform
sample space, then the probability of the event occurring is
# of elements in E
total number of possible outcomes in the sample space
.
Example 2: If three dice are thrown, find the probability that the sum of
the three dice is six.
◮
◮
First we find the size of the sample space: 6 × 6 × 6 = 216 by the
multiplication principle
Next we find all outcomes in our event:
{(1, 2, 3), (1, 3, 2), (1, 1, 4), (1, 4, 1), (2, 1, 3), (2, 3, 1), (2, 2, 2), (3, 1, 2), (3
Probability of an Event
◮
◮
A sample space is called uniform or equiprobable if each individual
possible outcome in the sample space has the same probability of
occurring as any other outcome.
If we have an event E (a subset of the sample space) in a uniform
sample space, then the probability of the event occurring is
# of elements in E
total number of possible outcomes in the sample space
.
Example 2: If three dice are thrown, find the probability that the sum of
the three dice is six.
◮
◮
◮
First we find the size of the sample space: 6 × 6 × 6 = 216 by the
multiplication principle
Next we find all outcomes in our event:
{(1, 2, 3), (1, 3, 2), (1, 1, 4), (1, 4, 1), (2, 1, 3), (2, 3, 1), (2, 2, 2), (3, 1, 2), (3
Since there are 10 possible outcomes in our event, the probability of
10
the event is 216
≈ 0.046, or about 4.6%.
Terminology from Genetics
◮
Genes are genetic material on a chromosome that code for a trait.
Terminology from Genetics
◮
Genes are genetic material on a chromosome that code for a trait. A
single trait is often determined by multiple genes, possibly on
different chromosomes.
Terminology from Genetics
◮
◮
Genes are genetic material on a chromosome that code for a trait. A
single trait is often determined by multiple genes, possibly on
different chromosomes.
An allele is a form of a gene at a single locus.
Terminology from Genetics
◮
◮
Genes are genetic material on a chromosome that code for a trait. A
single trait is often determined by multiple genes, possibly on
different chromosomes.
An allele is a form of a gene at a single locus. Example - human
have an eye color gene on chromosome 15 with two possible alleles:
B (brown) and b (blue).
Terminology from Genetics
◮
◮
◮
Genes are genetic material on a chromosome that code for a trait. A
single trait is often determined by multiple genes, possibly on
different chromosomes.
An allele is a form of a gene at a single locus. Example - human
have an eye color gene on chromosome 15 with two possible alleles:
B (brown) and b (blue).
Some alleles are dominant over others, and the nondominant alleles
are known as recessive alleles.
Terminology from Genetics
◮
◮
◮
◮
Genes are genetic material on a chromosome that code for a trait. A
single trait is often determined by multiple genes, possibly on
different chromosomes.
An allele is a form of a gene at a single locus. Example - human
have an eye color gene on chromosome 15 with two possible alleles:
B (brown) and b (blue).
Some alleles are dominant over others, and the nondominant alleles
are known as recessive alleles.
A genotype is the actual set of alleles an organism carries.
Terminology from Genetics
◮
◮
◮
◮
Genes are genetic material on a chromosome that code for a trait. A
single trait is often determined by multiple genes, possibly on
different chromosomes.
An allele is a form of a gene at a single locus. Example - human
have an eye color gene on chromosome 15 with two possible alleles:
B (brown) and b (blue).
Some alleles are dominant over others, and the nondominant alleles
are known as recessive alleles.
A genotype is the actual set of alleles an organism carries. Since
humans have pairs of chromosomes, each gene contains one or more
pairs of allels and may involve alleles at differnt loci.
Terminology from Genetics
◮
◮
◮
◮
◮
Genes are genetic material on a chromosome that code for a trait. A
single trait is often determined by multiple genes, possibly on
different chromosomes.
An allele is a form of a gene at a single locus. Example - human
have an eye color gene on chromosome 15 with two possible alleles:
B (brown) and b (blue).
Some alleles are dominant over others, and the nondominant alleles
are known as recessive alleles.
A genotype is the actual set of alleles an organism carries. Since
humans have pairs of chromosomes, each gene contains one or more
pairs of allels and may involve alleles at differnt loci.
The genotype is denoted as a pair of letters that represent the pair
of alleles for that particular gene.
Terminology continued...
◮
Example - Eye color gene at one locus has two alleles B and b, so
there are four possible genotypes: BB, Bb, bB, bb.
Terminology continued...
◮
◮
Example - Eye color gene at one locus has two alleles B and b, so
there are four possible genotypes: BB, Bb, bB, bb.
Genes with two dominant alleles are called homozygous dominant,
genes with two recessive alleles are homozygous recessive.
Terminology continued...
◮
◮
Example - Eye color gene at one locus has two alleles B and b, so
there are four possible genotypes: BB, Bb, bB, bb.
Genes with two dominant alleles are called homozygous dominant,
genes with two recessive alleles are homozygous recessive. Genes
with different alleles at a locus are heterozygous.
Terminology continued...
◮
◮
◮
Example - Eye color gene at one locus has two alleles B and b, so
there are four possible genotypes: BB, Bb, bB, bb.
Genes with two dominant alleles are called homozygous dominant,
genes with two recessive alleles are homozygous recessive. Genes
with different alleles at a locus are heterozygous.
A phenotype is the physical expression of a trait.
Terminology continued...
◮
◮
◮
◮
Example - Eye color gene at one locus has two alleles B and b, so
there are four possible genotypes: BB, Bb, bB, bb.
Genes with two dominant alleles are called homozygous dominant,
genes with two recessive alleles are homozygous recessive. Genes
with different alleles at a locus are heterozygous.
A phenotype is the physical expression of a trait.
A Punnett square is a diagram used to show the potential genotypes
resulting from a mating where the genotype of each parent is known.
Terminology continued...
◮
◮
◮
◮
Example - Eye color gene at one locus has two alleles B and b, so
there are four possible genotypes: BB, Bb, bB, bb.
Genes with two dominant alleles are called homozygous dominant,
genes with two recessive alleles are homozygous recessive. Genes
with different alleles at a locus are heterozygous.
A phenotype is the physical expression of a trait.
A Punnett square is a diagram used to show the potential genotypes
resulting from a mating where the genotype of each parent is known.
Examples
The gene determining albinism can have dominant allele A or recessive
allele a.
Examples
The gene determining albinism can have dominant allele A or recessive
allele a. Two parents both have genotype Aa. Find the probability that
their child will be (a) an albino, and (b) a carrier.
Examples
The gene determining albinism can have dominant allele A or recessive
allele a. Two parents both have genotype Aa. Find the probability that
their child will be (a) an albino, and (b) a carrier.
◮
We can draw the Punnett square:
Examples
The gene determining albinism can have dominant allele A or recessive
allele a. Two parents both have genotype Aa. Find the probability that
their child will be (a) an albino, and (b) a carrier.
◮
We can draw the Punnett square:
A
a
A
AA
aA
a
Aa
aa
Examples
The gene determining albinism can have dominant allele A or recessive
allele a. Two parents both have genotype Aa. Find the probability that
their child will be (a) an albino, and (b) a carrier.
◮
◮
A
a
A AA Aa
a aA aa
Assuming each outcome is equally likely, we see that the probability
the child will be albino is 41 .
We can draw the Punnett square:
Examples
The gene determining albinism can have dominant allele A or recessive
allele a. Two parents both have genotype Aa. Find the probability that
their child will be (a) an albino, and (b) a carrier.
◮
◮
◮
A
a
A AA Aa
a aA aa
Assuming each outcome is equally likely, we see that the probability
the child will be albino is 41 .
The probability they will be a carrier is 42 = 21 .
We can draw the Punnett square:
Blood Type
A father and mother have blood types AO and AB respectively. What is
the probability that their child has blood type (a) A, (b) B, (c) AB, (d)
O?
Blood Type
A father and mother have blood types AO and AB respectively. What is
the probability that their child has blood type (a) A, (b) B, (c) AB, (d)
O?
A
B
Our Punnett square is: A AA AB
O OA OB
Blood Type
A father and mother have blood types AO and AB respectively. What is
the probability that their child has blood type (a) A, (b) B, (c) AB, (d)
O?
A
B
Our Punnett square is: A AA AB
O OA OB
(a) Since O is recessive and both A and B are dominant,
Blood Type
A father and mother have blood types AO and AB respectively. What is
the probability that their child has blood type (a) A, (b) B, (c) AB, (d)
O?
A
B
Our Punnett square is: A AA AB
O OA OB
(a) Since O is recessive and both A and B are dominant,we see that the
blood type A corresponds to both AA and AO.
Blood Type
A father and mother have blood types AO and AB respectively. What is
the probability that their child has blood type (a) A, (b) B, (c) AB, (d)
O?
A
B
Our Punnett square is: A AA AB
O OA OB
(a) Since O is recessive and both A and B are dominant,we see that the
blood type A corresponds to both AA and AO.So the probability of
blood type A is 42 = 12 .
Blood Type
A father and mother have blood types AO and AB respectively. What is
the probability that their child has blood type (a) A, (b) B, (c) AB, (d)
O?
A
B
Our Punnett square is: A AA AB
O OA OB
(a) Since O is recessive and both A and B are dominant,we see that the
blood type A corresponds to both AA and AO.So the probability of
blood type A is 42 = 12 .
(b) Here, we have only OB yielding blood type B,
Blood Type
A father and mother have blood types AO and AB respectively. What is
the probability that their child has blood type (a) A, (b) B, (c) AB, (d)
O?
A
B
Our Punnett square is: A AA AB
O OA OB
(a) Since O is recessive and both A and B are dominant,we see that the
blood type A corresponds to both AA and AO.So the probability of
blood type A is 42 = 12 .
(b) Here, we have only OB yielding blood type B,so the probability of
blood type B is 14 .
Blood Type
A father and mother have blood types AO and AB respectively. What is
the probability that their child has blood type (a) A, (b) B, (c) AB, (d)
O?
A
B
Our Punnett square is: A AA AB
O OA OB
(a) Since O is recessive and both A and B are dominant,we see that the
blood type A corresponds to both AA and AO.So the probability of
blood type A is 42 = 12 .
(b) Here, we have only OB yielding blood type B,so the probability of
blood type B is 14 .
(c) For type AB, we again have the probability is 41 .
Blood Type
A father and mother have blood types AO and AB respectively. What is
the probability that their child has blood type (a) A, (b) B, (c) AB, (d)
O?
A
B
Our Punnett square is: A AA AB
O OA OB
(a) Since O is recessive and both A and B are dominant,we see that the
blood type A corresponds to both AA and AO.So the probability of
blood type A is 42 = 12 .
(b) Here, we have only OB yielding blood type B,so the probability of
blood type B is 14 .
(c) For type AB, we again have the probability is 41 .
(d) Since we have no occurances of OO for this couple,
Blood Type
A father and mother have blood types AO and AB respectively. What is
the probability that their child has blood type (a) A, (b) B, (c) AB, (d)
O?
A
B
Our Punnett square is: A AA AB
O OA OB
(a) Since O is recessive and both A and B are dominant,we see that the
blood type A corresponds to both AA and AO.So the probability of
blood type A is 42 = 12 .
(b) Here, we have only OB yielding blood type B,so the probability of
blood type B is 14 .
(c) For type AB, we again have the probability is 41 .
(d) Since we have no occurances of OO for this couple, it is impossible
for the child to have blood type O.
Blood Type
A father and mother have blood types AO and AB respectively. What is
the probability that their child has blood type (a) A, (b) B, (c) AB, (d)
O?
A
B
Our Punnett square is: A AA AB
O OA OB
(a) Since O is recessive and both A and B are dominant,we see that the
blood type A corresponds to both AA and AO.So the probability of
blood type A is 42 = 12 .
(b) Here, we have only OB yielding blood type B,so the probability of
blood type B is 14 .
(c) For type AB, we again have the probability is 41 .
(d) Since we have no occurances of OO for this couple, it is impossible
for the child to have blood type O.