lecture ii. using the example from birenens chapter 1: assume we are interested in the game texas...

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Lecture II

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Page 1: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Lecture II

Page 2: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto). In this game, players choose a set of 6

numbers out of the first 50. Note that the ordering does not count so that 35,20,15,1,5,45 is the same of 35,5,15,20,1,45.

How many different sets of numbers can be drawn? First, we note that we could draw any one of 50

numbers in the first draw.

Page 3: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

However for the second draw we can only draw 49 possible numbers (one of the numbers has been eliminated). Thus, there are 50 x 49 different ways to draw two numbers

Again, for the third draw, we only have 48 possible numbers left. Therefore, the total number of possible ways to choose 6 numbers out of 50 is

50

5 501

50 61 45

1

50!50

50 6 !k

j k

k

kj k

k

Page 4: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Finally, note that there are 6! ways to draw a set of 6 numbers (you could draw 35 first, or 20 first, …). Thus, the total number of ways to draw an unordered set of 6 numbers out of 50 is

This is a combinatorial. It also is useful for binomial arithmetic:

50 50!

15,890,7006 6! 50 6 !

0

nn k n k

k

na b a b

k

Page 5: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Basic definitions: Sample Space: The set of all possible

outcomes. In the Texas lotto scenario, the sample space is

all possible 15,890,700 sets of 6 numbers which could be drawn.

Event: A subset of the sample space. A subset of the sample space. In the Texas lotto

scenario, possible events include single draws such as {35,20,15,1,5,45} or complex draws such as all possible lotto tickets including {35,20,15}. Note that this could be {35,20,15,1,2,3}, {35,20,15,1,2,4},….

Page 6: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Simple Event: An event which cannot be a union of other events An event which cannot be a union of other

events. In the Texas lotto scenario, this is a single draw such as {35,20,15,1,5,45}.

Composite Event: An event which is not a simple event.

Page 7: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

A set

of different combinations of outcomes is called an event. These events could be simple events or compound events. In the Texas lotto case, the important aspect is that the event is something you could bet on (for example, you could bet on three numbers in the draw {35,20,15}).

1,

kj j

Page 8: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

A collection of events F is called a family of subsets of sample space Ω. This family consists of all possible subsets of Ω including Ω itself and the null-set Φ. Following the betting line, you could

bet on all possible numbers (covering the board) so that Ω is a valid bet.

Alternatively, you could bet on nothing, or Φ is a valid bet.

Page 9: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Next, we will examine a variety of closure conditions. These are conditions that guarantee that if one set is an contained in a family, another related set must also be contained in that family.First, we note that the family is

closed under complementarity

If then \A F A A F

Page 10: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Second, we note that the family is closed under union

Definition 1.1 (Bierens): A collection F of subsets of a nonempty set Ω satisfying closure under complementarity and closure under union is called an algebra.

Adding closure under infinite union defined as

If , then A B F A B F

1

If for 1,2,3, then j jj

A F j A F

Page 11: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Definition 1.2 (Bierens): A collection F of subsets of a nonempty set Ω satisfying closure under complementarity and infinite union is called a σ-algebra (sigma-algebra) or a Borel Field.

Page 12: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

A

P A

0,1x

Page 13: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

We typically think of this as an odds function (i.e., what are the odds of a winning lotto ticket? 1/15,890,700).To be mathematically precise, suppose

we define a set of events

say that we choose n different numbers. The probability of winning the lotto is

1, jA

1, nP A P n N

Page 14: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Our intuition would indicate that ,

or the probability of winning given that you have covered the board is equal to one (a certainty).

Further, if you don’t bet the probability of winning is zeros or

1P

0P

Page 15: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Definition 1.2.2 (Cassella and Berger) : Given a sample space Ω and an associated Borel field B, a probability function is a function P with domain B that satisfies P (A)0 for all AB. P (Ω)=1. If A1,A2,…B are pairwise disjoint, then

P(i=1Ai)=

i=1P (Ai)

Page 16: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Axioms of Probability: P(A) 0 for any event A. P(S) = 1 where S is the sample space. If {Ai}, i=1,2,…, are mutually exclusive

(that is, AiAj= for all ij), then P(A1A2…)=P(A1)+P(A2)+…

Page 17: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

In a little more detail from Casella and Berger: Definition 1.1.1: The set, S, of all possible

outcomes of a particular experiment is called the sample space for the experiment.

Definition 1.1.2: An event is any collection of possible outcomes of an experiment, that is, any subset of S (including S itself).

Page 18: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Defining the subset relationship A B x A xB A = B A B and B A Union: The union of A and B, written A B, is the

set of elements that belong to either A or B.

Intersection: The intersection of A and B, written A B, is the set of elements that belong to both A and B.

: and A B x x A x B

: or A B x x A x B

Page 19: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Complementation: The complement of A, written Ac, is the set of all elements that are not in A.

:cA x x A

Page 20: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Theorem 1.1.1: For any three events A, B, and C defined on a sample space S, Commutativity: A B=B A, A B=B A. Associativity: A (B C)=(A B) C,

A (B C)=(A B) C Distributative Laws:

A (B C )=(A B )(A C ),A (B C )=(A B )(A C )

DeMorgan’s Laws: (A B )c=Ac Bc, (A B )c=Ac Bc

Page 21: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Simple Evens with Equal Probabilities

the probability of event A is simply the

number possible occurrences of A divided by the number of possible occurrences in the sample.

)(

)()(

Sn

AnAP

Page 22: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Definition 2.3.1 The number of permutations of taking r elements from n elements is a number of distinct ordered sets consisting of r distinct elements which can be formed out of a set of n distinctive elements and is denoted Pn

r.

Page 23: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

The first point to consider is that of factorials. For example, if you have two objects A and B, how many different ways are there to order the object? Two:

{A, B} or {B, A} If you have three orderings how many ways

are there to order the objects? Six:{A, B, C}, {A, C, B}, {B, A, C}, {B, C, A}, {C, A,

B}, or {C, B, A}

Page 24: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

The sequence then becomes two objects can be drawn in two sequences, three objects can be drawn in six sequences (2 x 3). By inductive proof, four objects can be drawn in 24 sequences (6 x 4).

The total possible number of sequences is then for n objects is n! defined as:

n!=n (n -1)(n -2)…1

Page 25: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Theorem 2.3.1: Pnr=n!/(n-r)!.

Definition 2.3.2: The number of combinations of taking r elements from n elements is the number of distinct sets consisting of r distinct elements which can be formed out of a set of n distinct elements and is denoted Cn

r.

!)!(

!

rrn

nC nr

Page 26: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

In order to define the concept of a conditional probability it is necessary to discuss joint probabilities and marginal probabilities. A joint probability is the probability of two

random events. For example, the draw of two cards from a deck of cards. There are 52x51=2652 different combinations of the first two cards from the deck.

Page 27: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

The marginal probability is overall probability of a single event or the probability of drawing a given card.

The conditional probability of an event is the probability of that event given that some other event has occurred. In the textbook, what is the probability of the die

being a one if you know that the face number is odd? (1/3).

Note if you know that the role of the die is a one, then the probability of the role being odd is 1.

Page 28: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Axioms of Conditional Probability: P (A|B )0 for any event A. P (A|B )=1 for any event A B. If {AiB}, i=1, 2,… are mutually exclusive,

then P(A1A2…|B )=P(A1|B )+P(A2|B)+…. If B H and B G and P (G )0, then

)(

)(

)|(

)|(

GP

HP

BGP

BHP

Page 29: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Theorem 2.4.1: P (A|B )=P (AB )/P (B) for any pair of events A and B such that P (B)>0.

Theorem 2.4.2 (Bayes Theorem): Let Events A1, A2, …, An be mutually exclusive such that P (A1A2…An)=1 and P (Ai) >0 for each i. Let E be an arbitrary event such that P (E)>0. Then

n

jjj

iii

APAEP

APAEPEAP

1

)()|(

)()|()|(

Page 30: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Another manifestation of this theorem is from the joint distribution function:

The bottom equality reduces the marginal probability of event E

( , )i i i iP E A P E A P E A P A

n

iii APAEPEP

1

)()|()(

Page 31: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

This yields a friendlier version of Bayes theorem based on the ratio between the joint and marginal distribution function:

)(

),()|(

EP

AEPEAP i

i

Page 32: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Statistical independence is when the probability of one random variable is independent of the probability of another random variable.

Definition 2.4.1: Events A and B are said to be independent if P (A)=P (A|B ).

Page 33: Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,

Definition 2.4.2: Events A, B, and C are said to be mutually independent if the following equalities hold: P (AB )=P (A )P (B ) P (AC )=P (A )P (C ) P (BC )=P (B )P (C ) P (AB C )=P (A )P (B)P (C )