Chapter 7 - Part Two Counting Techniques Wednesday, March 18, 2009.

Slides:

Advertisements

Similar presentations

A binomial is a polynomial with two terms such as x + a. Often we need to raise a binomial to a power. In this section we'll explore a way to do just.

Advertisements

Permutations and Combinations

Beginning Probability

MATHCOUNTS TOOLBOX Facts, Formulas and Tricks

Consider the possible arrangements of the letters a, b, and c. List the outcomes in the sample space. If the order is important, then each arrangement.

Statistics Review. Box-and-Whisker Plots The Parts of a Box and Whisker Plot Name the parts of a Box-and-Whisker Plot MedianUpper Quartile Lower.

Section 3-6 Counting. FUNDAMENTAL COUNTING RULE For a sequence of two events in which the first event can occur m ways and the second event can occur.

Section 4-6 Counting.

How many possible outcomes can you make with the accessories?

12.1 & 12.2: The Fundamental Counting Principal, Permutations, & Combinations.

1 Counting Rules. 2 The probability of a specific event or outcome is a fraction. In the numerator we have the number of ways the specific event can occur.

Chapter 11 Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved.

Permutations and Combinations AII Objectives:  apply fundamental counting principle  compute permutations  compute combinations  distinguish.

Expected value a weighted average of all possible values where the weights are the probabilities of each outcome :

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.

Math 2 Honors - Santowski

Many Experiments can be done with the results of each trial reduced to 2 outcomes Binomial Experiment: There are n independent trials Each trial has only.

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.

Topics to be covered: Produce all combinations and permutations of sets. Calculate the number of combinations and permutations of sets of m items taken.

Counting Techniques 0.4.

Permutations and Combinations. Objectives:  apply fundamental counting principle  compute permutations  compute combinations  distinguish permutations.

Section 10-3 Using Permutations and Combinations.

© The McGraw-Hill Companies, Inc., Chapter 4 Counting Techniques.

Part 2 – Factorial and other Counting Rules

Binomial Probability Distribution

Homework Homework due now. Reading: relations

Permutations and Combinations

PERMUTATIONS AND COMBINATIONS BOTH PERMUTATIONS AND COMBINATIONS USE A COUNTING METHOD CALLED FACTORIAL.

Permutations and Combinations. Objectives:  apply fundamental counting principle  compute permutations  compute combinations  distinguish permutations.

Copyright © Ed2Net Learning Inc.1. 2 Warm Up Tell whether each event is dependent or independent 1. Selecting a name from a Chicago telephone book and.

Lesson 0.4 (Counting Techniques)

Probability and Counting Rules

Counting Techniques Tree Diagram Multiplication Rule Permutations Combinations.

I CAN: Use Permutations and Combinations

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

7.3 Combinations Math A combination is a selection of a group of objects taken from a larger pool for which the kinds of objects selected is of.

Permutations and Combinations

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

Binomial Distribution. Bernoulli Trials Repeated identical trials are called Bernoulli trials if: 1. There are two possible outcomes for each trial, denoted.

Permutations and Combinations AII Objectives:  apply fundamental counting principle  compute permutations  compute combinations  distinguish.

Quiz: Draw the unit circle: Include: (1)All “nice” angles in degrees (2) All “nice” angles in radians (3) The (x, y) pairs for each point on the unit circle.

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

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.

Permutations and Combinations. Permutations Definition –An ordered arrangement of objects is called a permutation. –Hence, a permutation of n distinct.

Permutations and Combinations

Chapter 10 Counting Methods.

4-1 Chapter 4 Counting Techniques.

Apply the Counting Principle and Permutations

Chapter 0.4 Counting Techniques.

4-1 Chapter 4 Counting Techniques.

4-1 Chapter 4 Counting Techniques.

Warm Up Permutations and Combinations Evaluate  4  3  2  1

Probability: Living with the Odds

Permutations and Combinations

Permutations and Combinations

Lesson 11-1 Permutations and Combinations

Permutations and Combinations

Probability: Living with the Odds

Permutations and Combinations

How many possible outcomes can you make with the accessories?

4-1 Chapter 4 Counting Techniques.

Bellwork Practice Packet 10.3 B side #3.

Permutations and Combinations

Standard DA-5.2 Objective: Apply permutations and combinations to find the number of possibilities of an outcome.

Permutations and Combinations

Lecture 7: Permutations and Combinations

Permutations and Combinations

Presentation transcript:

Chapter 7 - Part Two Counting Techniques Wednesday, March 18, 2009

Permutations:  A permutation of r elements from a set of n elements in any specific ordering or arrangement, without repetition of the r elements. Each arrangement is a different permutation.  Clue words: arrangement, schedule, order,....  Example: There are six permutations of the letters A, B, and C.  ABC ACB BAC BCA CBA CAB

Permutations Formula

Example  In the Olympics Gymnastics competition, 8 gymnasts compete for medals. How many ways can the medals be awarded (gold, silver, and bronze)?

Distinguishable Permutations  Objects are not all distinguishable, namely n 1 of type 1, n 2 of type 2, etc. The number of permutations is:

Example  How many permutations are there of the letters in the word STATISTICS?

Combinations  A subset of items selected without regard to order.  Clue words: group, committee, sample....  Example: There is only one combination of the letters A, B, and C === ABC

Combinations Formula

Example  How many committees of three people can be selected from a group of 8 people?

Pascal’s Triangle  Can be used to compute combinations

Baseball  How many ways can three outfielders and four infielders be chosen from five outfielders and seven infielders?

Lottery  In the Pennsylvania lottery drawing, 5 numbered balls are selected from a box containing balls numbered 1 through 40. How many different combinations of winning numbers are there?

Some Card Problems  I am playing a hand of 5 card poker. What is the probability that I am dealt the following: 3 Kings and 2 Aces? All hearts Exactly two aces Three of a kind.

Example  A barrel contains 15 apples. Of the apples, 5 are rotten and 10 are good. Three apples are selected at random. What is the probability of selecting at least one good apple?

Binomial Probability  Same experiment is repeated several times.  Only two possible outcomes: success and failure  Repeated trials are independent.  n = number of trials  x = number of successes  p = probability of success on each trial

Formula

Example  Flip a coin 20 times. What is the probability of getting 6 tails?

Example  I am taking a 10 question, multiple choice exam and I have not studied. Each question has 4 possible answers. By guessing only, what is the probability that I can get 6 questions correct?

Problem of the Day  A customer walks into a hardware store to buy something and asks the clerk how much 1 would cost and the clerk answers $1. The customer then asks how much 10 would cost and the clerk answers $2. The customer says, “I’ll buy 1515,” and pays the clerk $4. What was the customer buying?