Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics ofChance

November 6, 2013

Chapter 8: Probability: The Mathematics of Chance

Last Time

Crystallographic notation

Groups


Crystallographic notation

The first symbol is always a p, which indicates that thepattern repeats (is “periodic”) in the horizontal direction.

The second symbol is m if there is a vertical line of reflection.Otherwise, it is 1.

The third symbol is

m (for “mirror”), if there is a horizontal line of reflection (inwhich case there is also glide reflection)a (for “alternating”), if there is a glide reflection but nohorizontal reflection1 if there is no horizontal reflection or glide reflection

The fourth symbol is 2, if there is half-turn rotationalsymmetry; otherwise, it is 1.


Group

A group is a collection of elements {A,B, · · · } and an operation ◦between pairs of them such that the following properties hold:

Closure: The result of one element operating on another isitself an element of the collection (A ◦ B is in the collection).

Identity element: There is a special element I , called theidentity element, such that the result of an operation involvingthe identity and any element is that same element (I ◦ A = Aand A ◦ I = A).

Inverses: For any element A, there is another element, calledits inverse and denoted A−1, such that the result of anoperation involving an element and its inverse is the identityelement (A ◦ A−1 = I and A−1 ◦ A = I ).

Associativity: The result of several consecutive operations isthe same regardless of grouping or parenthesizing, providedthat the consecutive order of operations is maintained:A ◦ B ◦ C = A ◦ (B ◦ C ) = (A ◦ B) ◦ C .


This Time

Probability Models and Rules

Discrete Probability Models

Equally Likely Outcomes


Probability

Random

A phenomenon or trial is said to be random if individual outcomesare uncertain but the long-term pattern of many individualoutcomes is predictable.

Probability

The probability of any outcome of a random phenomenon is theproportion of times the outcome would occur in a very long seriesof repetitions. Probabilities can be expressed as decimals,percentages, or fractions.


Probability Model

Sample Space

The sample space S of a random phenomenon is the set of allpossible outcomes that cannot be broken down further into simplercomponents.

Event

An event is any outcome or any set of outcomes of a randomphenomenon. That is, an event is a subset of the sample space.

Probability Model

A probability model is a mathematical description of a randomphenomenon consisting of two pairs: a sample space S and a wayof assigning probabilities to events.


Example

Consider a simple coin tossThe sample space is {H,T}, with two events, each withprobability 1

2Consider n simple coin tossesThe sample space consists of all 2n outcomes each with probability12n


Question

What is the probability that two dice rolled will sum to 7? sum to5?


Events

Complement of an Event

The complement of an event A is the event that A does notoccur, written as AC .

Disjoints Events

Two events are disjoint events if they have no outcome incommon. Disjoint events are also called mutually exclusive events.

Independent Events

Two events are independent events if the occurrence of oneevent has no influence on the probability of the occurrence of theother event.


Probability Rules

Probability Rules

Rule 1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1.

Rule 2. If S is the sample space in a probability model, then P(S) = 1.

Rule 3. The complement rule: P(AC ) = 1− P(A).

Rule 4. The multiplication rule for independent events:P(A and B) = P(A)× P(B).

Rule 5. The general addition rule:P(A or B) = P(A) + P(B)− P(A and B).

Rule 6. The addition rule for disjoint events:P(A or B) = P(A) + P(B).


Discrete Probability Models

Discrete Probability Model

A discrete probability model is a probability model with acountable number of outcomes in its sample space.

Example: Dice rollCoin flip


Equally Likely Outcome

Finding Probabilities of Equally Likely Outcomes

If a random phenomenon has equally likely outcomes, then theprobability of event A is

P(A) =count of outcomes in event A

count of outcomes in sample space S


Combinatorics

Combinatorics is the study of methods for counting.

Permutation

A permutation is an ordered arrangement of k items that arechosen without replacement from a collection of n items. It can benotated as P(n, k), nPk of Pn

k and has the formula

P(n, k) = n × (n − 1)× · · · × (n − k + 1)

Factorial

The factorial for a positive integer n equals the product of thefirst n positive integers. The term “n factorial” is notated n!:

n × (n − 1)× (n − 2)× · · · × 3× 2× 1.

By convention, we define 0! to equal 1, which can be interpretedas saying there is one way to arrange zero items.


Combinations

Factorial

The factorial for a positive integer n equals the product of thefirst n positive integers. The term “n factorial” is notated n!:

n × (n − 1)× (n − 2)× · · · × 3× 2× 1.

By convention, we define 0! to equal 1, which can be interpretedas saying there is one way to arrange zero items.

Combination

A combination is an unordered arrangement of k items that arechosen without replacement from a collection of n items. It can benotated as

(nk

), C (n, k), or nCk and is sometimes spoken “n

choose k”.

C (n, k) =n × (n − 1)× · · · × (n − k + 1)

k!=

n!

k!(n − k)!


Counting distinct items

Counting Ordered Collections of Distinct Items

Rule A. Suppose we have a collection of n distinct items. Wewant to arrange k of these items in order, and the same item canappear more than once in the arrangement. The number ofpossible arrangements is

n × n × · · · × n = nk

Rule B. (Permutations) Suppose we have a collection of ndistinct items. We want to arrange k of these items in order, andany item can appear no more than once in the arrangement. Thenumber of possible arrangements is

n × (n − 1)× · · · × (n − k + 1)


Counting Distinct Items

Counting Unordered Collections of Distinct Items

Rule C. Suppose that we have a collection of n distinct items. Wewant to select k of those items with no regard to order, and anyitem can appear more than once in the collection. The number ofpossible collections is

(n + k − 1)!

k!(n − 1)!

Rule D. (Combinations) Suppose that we have a collection of ndistinct items. We want to select k of these items with no regardto order, and any item can appear no more than once in thecollection. The number of possible selections is

n!

k!(n − k)!


Question

Choose a young adult (aged 25 to 34 years) at random. Theprobability is 0.12 that the person choose did not complete highschool, 0.31 that the person has a high school diploma but nofurther education, and 0.29 that the person has at least abachelor’s degree.

(a) What must be the probability that a randomly chosen youngadult has some education beyond high school but does nothave a bachelor’s degree?

(b) What is the probability that a randomly chosen young adulthas at least a high school education?


Question

You toss a balanced coin 10 times and write down the resultingsequence of heads and tails.

(a) How many possible outcomes are there for 10 tosses?

(b) What is the probability that your 10-toss sequence is either allheads or all tails?

(c) What is the probability that your 10-toss sequence has acombine 5 heads and a combine 5 tails?

In poker, a royal flush is a 5-card hand containing an ace, king,queen, jack, and 10, all of the same suit.

(a) How many royal flush hands are possible?

(b) What is the number of 5-card hands possible from a 52-carddeck?

(c) What is the probability that 5 cards drawn at random from a52-card deck will yield a royal flush?


Question

Suppose a monkey is at a type writer and can only press a, r , e.

1 How many possible three-letter “words” can the monkey typeusing only these letters?

2 Which of these are words in an English dictionary?

3 What is the probability that the word the monkey typed is in aEnglish dictionary?

4 What if the word can be any length?


Next time

Quiz over chapter 19 and chapter 8

Continuous Probability Models


Chapter 8: Probability: The Mathematics of Chance

Documents

Transcript of Chapter 8: Probability: The Mathematics of Chance