Chapter 8 Counting Principles: Further Probability Topics Section 8.3 Probability Applications of Counting Principles.

Slides:

Advertisements

Similar presentations

Discrete Probability Rosen 5.1.

Advertisements

Probability – Page 1CSCI 1900 – Discrete Structures CSCI 1900 Discrete Structures Probability Reading: Kolman, Section 3.4.

Thinking Mathematically

16.4 Probability Problems Solved with Combinations.

5.4 Counting Methods Objectives: By the end of this section, I will be able to… 1) Apply the Multiplication Rule for Counting to solve certain counting.

How many possible outcomes can you make with the accessories?

Chapter 3b Counting Rules. Permutations  How many ways can 5 students in a class of 30 be assigned to the front row in the seating chart?  There are.

Today Today: Reading: –Read Chapter 1 by next Tuesday –Suggested problems (not to be handed in): 1.1, 1.2, 1.8, 1.10, 1.16, 1.20, 1.24, 1.28.

1 Probability Parts of life are uncertain. Using notions of probability provide a way to deal with the uncertainty.

4/5/05Tucker, Sec Applied Combinatorics, 4rth Ed. Alan Tucker Section 5.4 Distributions Prepared by Jo Ellis-Monaghan.

Basics of Probability. Trial or Experiment Experiment - a process that results in a particular outcome or “event”. Simple event (or sample point), E i.

1 Section 5.1 Discrete Probability. 2 LaPlace’s definition of probability Number of successful outcomes divided by the number of possible outcomes This.

Probability Using Permutations and Combinations

1 The game of poker You are given 5 cards (this is 5-card stud poker) The goal is to obtain the best hand you can The possible poker hands are (in increasing.

College Algebra Fifth Edition

Permutations and Combinations. Random Things to Know.

4. Counting 4.1 The Basic of Counting Basic Counting Principles Example 1 suppose that either a member of the faculty or a student in the department is.

Chapter 8 Counting Principles: Further Probability Topics Section 8.2 Combinations.

NA387 Lecture 5: Combinatorics, Conditional Probability

X of Z: MAJOR LEAGUE BASEBALL ATTENDANCE Rather than solving for z score first, we may be given a percentage, then we find the z score, then we find the.

Formulas and Principles. Math I Unit 4 If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events.

Probability – Models for random phenomena

Chapter 11: Counting Methods

Counting. Techniques for counting Rule 1 Suppose we carry out have a sets A 1, A 2, A 3, … and that any pair are mutually exclusive (i.e. A 1  A 2 =

3.8 Counting Techniques: Combinations. If you are dealt a hand in poker (5 cards), does it matter in which order the cards are dealt to you? A  K  J.

EGR Sample Space, S The set of all possible outcomes of an experiment. Each outcome is an element or member or sample point. If the set is finite.

Finding Probability Using Tree Diagrams and Outcome Tables

6.1 Probability exercises. Def Def: If S is a finite sample space of equally likely outcomes, and E is an event, that is, a subset of S, then the probability.

College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson.

Chapter 7 With Question/Answer Animations. Section 7.1.

March 10,  Counting  Fundamental Counting principle  Factorials  Permutations and combinations  Probability  Complementary events  Compound.

Basic Probability Section 7.1. Definitions Sample Space: The set of all possible outcomes in a given experiment or situation Often denoted by S Event:

1 Discrete Structures – CNS2300 Text Discrete Mathematics and Its Applications (5 th Edition) Kenneth H. Rosen Chapter 5 Counting.

Homework Homework due now. Reading: relations

Section 11.4 Tree Diagrams, Tables, and Sample Spaces Math in Our World.

Sec 4.4 Counting Rules Bluman, Chapter 4 A Question to Ponder:  A box contains 3 red chips, 2 blue chips and 5 green chips. A chip is selected, replaced.

Ch Counting Principles. Example 1  Eight pieces of paper are numbered from 1-8 and placed in a box. One piece of paper is drawn from the box, its.

ICS 253: Discrete Structures I Discrete Probability King Fahd University of Petroleum & Minerals Information & Computer Science Department.

 m C r =m!/r!(m-r)!  Property: m C r = m C (m-r)  Example: ◦ If you have 10 pens, all in different colors, and you want to bring two of them to class,

Probability What are the chances?.

Some other set notation We will use the notation to mean that e is an element of A. We will use the notation to mean that e is not an element of A.

Probability and Counting Rules

37. Permutations and Combinations. Fundamental Counting Principle Fundamental Counting Principle states that if an event has m possible outcomes and another.

 Counting  Fundamental Counting principle  Factorials  Permutations and combinations  Probability  Complementary events  Compound events  Independent.

8.6 Counting Principles. Listing Possibilities: Ex 1 Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of paper is drawn from.

1 Discrete Structures - CSIS2070 Text Discrete Mathematics and Its Applications Kenneth H. Rosen Chapter 4 Counting.

COUNTING Permutations and Combinations. 2Barnett/Ziegler/Byleen College Mathematics 12e Learning Objectives for Permutations and Combinations  The student.

MATH 2311 Section 2.3. Basic Probability Models The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur.

Counting Techniques (Dr. Monticino). Overview  Why counting?  Counting techniques  Multiplication principle  Permutation  Combination  Examples.

Sample Spaces and Probability Addition Rules Multiplication Rules and Conditional Probability Counting Rules Probability and Counting Rules

Probability and statistics - overview Introduction Basics of probability theory Events, probability, different types of probability Random variable, probability.

Introduction to probability (3) Definition: - The probability of an event A is the sum of the weights of all sample point in A therefore If A1,A2,…..,An.

1 What Is Probability?. 2 To discuss probability, let’s begin by defining some terms. An experiment is a process, such as tossing a coin, that gives definite.

13 Lesson 1 Let Me Count the Ways Fundamental Counting Principle, Permutations & Combinations CP Probability and Statistics FA 2014 S-ID.1S-CP.3S-CP.5.

Warm up How many ways can 8 children be placed on a 8- horse Merry-Go-Round? 7! = What if Simone insisted on riding the red horse? Here we are only.

Counting Principles Ex. Eight pieces of paper are numbered 1 to 8 and placed in a box. One piece of paper is drawn from the box, its number is written.

Counting Unit 1.

What Is Probability?.

Sequences, Series, and Probability

Chapter 0.4 Counting Techniques.

8.3 Counting Apply the fundamental counting principle

4 Probability Lesson 4.8 Combinations and Probability.

Permutations and Combinations

Combination and Permutations Quiz!

Probability Section 19 NOTE: We are skipping Section 18. 2/21/2019

How many possible outcomes can you make with the accessories?

Bellwork Practice Packet 10.3 B side #3.

Applied Combinatorics, 4th Ed. Alan Tucker

Chapter 11: Further Topics in Algebra

Presentation transcript:

Chapter 8 Counting Principles: Further Probability Topics Section 8.3 Probability Applications of Counting Principles

Recall that tree diagrams can be used to solve probability problems involving dependent events, but tree diagrams sometimes require a large number of branches. Recall that tree diagrams can be used to solve probability problems involving dependent events, but tree diagrams sometimes require a large number of branches. Permutations and combinations are especially helpful when the numbers involved are large. Permutations and combinations are especially helpful when the numbers involved are large.

The use of combinations to solve probability problems depends on the basic probability principle: The use of combinations to solve probability problems depends on the basic probability principle: Let S be a sample space with equally likely outcomes, and let event E be a subset of S. Then the probability that event E occurs, written P(E), is where n(E) and n(S) represent the number of elements in sets E and S.

Combinations tend to be used more often than permutations in probability because many times the final result does not depend on the order of events. Combinations tend to be used more often than permutations in probability because many times the final result does not depend on the order of events. However, order does matter in some cases, and permutations would need to be used. However, order does matter in some cases, and permutations would need to be used. In some cases, a mixture of permutations and combinations are needed to solve the problem. In some cases, a mixture of permutations and combinations are needed to solve the problem.

Example: You and your friends order twelve burritos to go from a Mexican restaurant, five with hot peppers and seven without; however, the restaurant forgot to label them. If you pick three burritos at random, what is the probability that all three have hot peppers? Example: You and your friends order twelve burritos to go from a Mexican restaurant, five with hot peppers and seven without; however, the restaurant forgot to label them. If you pick three burritos at random, what is the probability that all three have hot peppers? Find the number of ways you can choose 3 burritos from C 3 = 220 This is your sample space. 12 C 3 = 220 This is your sample space. Find the number of ways you can choose 3 burritos with hot peppers. 5 C 3 7 C 0 = 10 P(3 w/hot peppers) = __10__ =

Example: A basketball coach was asked to rank order his top three players from the starting five players. Suppose one of the players is named Jack. Example: A basketball coach was asked to rank order his top three players from the starting five players. Suppose one of the players is named Jack. a.) What is the probability that Jack is one of the top three players? three players? Find the number of ways the top 3 players can be chosen. 5 C 3 = 10 This is your sample space. Find the number of ways that Jack could be chosen as one of the three ranked players. 4 C 2 = 6 Since we want Jack to be one of the three chosen players, we only need to find out how many ways the coach can select the other two from the remaining 4 players. Determine the probability Jack is one of the top three. P(top three) = 6 =

Example: A basketball coach was asked to rank order his top three players from the starting five players. Suppose one of the players is named Jack. Example: A basketball coach was asked to rank order his top three players from the starting five players. Suppose one of the players is named Jack. b.) What is the probability that Jack is ranked # 1? Find the number of ways the top 3 players can be ranked. 5 P 3 = 60 This is your sample space. Find the number of ways that Jack could be chosen as the number one ranked player. _1_ 4 3 = 12 Since we want Jack to be the number one ranked player, we only need to find out how many ways the coach can select the second and third ranked players from the remaining 4. Determine the probability Jack is the top-ranked player. P( rank #1 ) = 12 =

Example: A standard poker hand consists of 5 cards. Find the probability that a person is dealt a straight flush (5 in a row in a single suit, but not a royal flush). Assume Aces are either high or low. Example: A standard poker hand consists of 5 cards. Find the probability that a person is dealt a straight flush (5 in a row in a single suit, but not a royal flush). Assume Aces are either high or low. A flush could start with an Ace, 2, 3, 4, 5, 6, 7, 8, or 9. This gives 9 choices in each of 4 suits, so there are 36 choices in all. ( 9 4 = 36 ) P(straight flush) = ___36____ 52 C 5 36______ 2, 598, 960 = ≈ = x 10 -5

Example: Two hundred people apply for three jobs. Sixty of the applicants are women. If three persons are selected at random, what is the probability that: Example: Two hundred people apply for three jobs. Sixty of the applicants are women. If three persons are selected at random, what is the probability that: a.) all are women? a.) all are women? b.) exactly two are men? c.) at least one is female?