Probability and Counting Exercises

Slides:

Advertisements

Similar presentations

Application: The Pigeonhole Principle Lecture 37 Section 7.3 Wed, Apr 4, 2007.

Advertisements

The Pigeonhole Principle

EXAMPLE 3 Find a probability using permutations For a town parade, you will ride on a float with your soccer team. There are 12 floats in the parade, and.

Finding a Binomial Probability

Elementary Number Theory and Methods of Proof

Chapter 2 Probability. 2.1 Sample Spaces and Events.

1 Counting. 2 Situations where counting techniques are used  You toss a pair of dice in a casino game. You win if the numbers showing face up have a.

Counting Techniques: Combinations

Counting and Probability The outcome of a random process is sure to occur, but impossible to predict. Examples: fair coin tossing, rolling a pair of dice,

Multiplication Rule. A tree structure is a useful tool for keeping systematic track of all possibilities in situations in which events happen in order.

Discrete Structures Chapter 4 Counting and Probability Nurul Amelina Nasharuddin Multimedia Department.

Chapter 4: Probability (Cont.) In this handout: Total probability rule Bayes’ rule Random sampling from finite population Rule of combinations.

Combinations We should use permutation where order matters

Discrete Mathematics Lecture 6 Alexander Bukharovich New York University.

1 More Counting Techniques Possibility trees Multiplication rule Permutations Combinations.

Copyright © Cengage Learning. All rights reserved. CHAPTER 9 COUNTING AND PROBABILITY.

POPULATION- the entire group of individuals that we want information about SAMPLE- the part of the population that we actually examine in order to gather.

Divisibility Rules and Mental Math

Functions A B f( ) =. This Lecture We will define a function formally, and then in the next lecture we will use this concept in counting. We will also.

Exploring Counting Rules & Probability Counting Aim: Find the number of possibilities.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 2 Probability.

Combinatorics 3/15 and 3/ Counting A restaurant offers the following menu: Main CourseVegetablesBeverage BeefPotatoesMilk HamGreen BeansCoffee.

In this chapter we will consider two very specific random variables where the random event that produces them will be selecting a random sample and analyzing.

6.2 Find Probability Using Permutations. Vocabulary n factorial: product of integers from 1 to n, written as n! 0! = 1 Permutation: arrangement of objects.

1 Permutations and Combinations. 2 In this section, techniques will be introduced for counting the unordered selections of distinct objects and the ordered.

Transparency 3 Click the mouse button or press the Space Bar to display the answers.

Counting. Product Rule Example Sum Rule Pigeonhole principle If there are more pigeons than pigeonholes, then there must be at least one pigeonhole.

PROBABILITY. FACTORIALS, PERMUTATIONS AND COMBINATIONS.

Counting and Probability. Counting Elements of Sets Theorem. The Inclusion/Exclusion Rule for Two or Three Sets If A, B, and C are finite sets, then N(A.

9.3 Addition Rule. The basic rule underlying the calculation of the number of elements in a union or difference or intersection is the addition rule.

Math 71B 11.1 – Sequences and Summation Notation 1.

More Combinations © Christine Crisp “Teach A Level Maths” Statistics 1.

Dr. Eng. Farag Elnagahy Office Phone: King ABDUL AZIZ University Faculty Of Computing and Information Technology CPCS 222.

Combination and Permutation. Example 1 There are 10 electional courses and a student has to take 6 of them. If 3 of 10 lessons are at the same time then,

Math Oct.29 th Prime numbers and composite numbers.

Arranging and Choosing © Christine Crisp “Teach A Level Maths” Statistics 1.

Permutations and Combinations Section 2.2 & 2.3 Finite Math.

Confidence Intervals (Dr. Monticino). Assignment Sheet  Read Chapter 21  Assignment # 14 (Due Monday May 2 nd )  Chapter 21 Exercise Set A: 1,2,3,7.

Discrete Structures Counting (Ch. 6)

Random Variables Example:

Unit VI Discrete Structures Permutations and Combinations SE (Comp.Engg.)

6.7 Permutations & Combinations. Factorial: 4! = 4*3*2*1 On calculator: math ==> PRB ==> 4 7! = 5040 Try 12!

11.1A Fundamental Counting Principal and Factorial Notation 11.1A Fundamental Counting Principal If a task is made up of multiple operations (activities.

Let A and B be two independent events for which P(A) = 0.15 and P(B) = 0.3. Find P(A and B).

Spring 2016 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University.

Divisibility and Mental Math Lesson 3-1. Vocabulary A number is divisible by another number if it can be divided into and result in a remainder of 0.

Special Topics. Calculating Outcomes for Equally Likely Events If a random phenomenon has equally likely outcomes, then the probability of event A is:

Microsoft produces a New operating system on a disk. There is 0

Discrete Mathematics. Exercises Exercise 1:  There are 18 Computer Science (CS) majors and 325 Business Administration (BA) majors at a college.

Spring 2016 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University.

Introduction to Probability Honors Geometry Summer School.

Counting and Probability. Imagine tossing two coins and observing whether 0, 1, or 2 heads are obtained. Below are the results after 50 tosses Tossing.

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

SECTION 5.4 COUNTING. Objectives 1. Count the number of ways a sequence of operations can be performed 2. Count the number of permutations 3. Count the.

CS336 F07 Counting 2. Example Consider integers in the set {1, 2, 3, …, 1000}. How many are divisible by either 4 or 10?

Discrete Mathematics Lecture 8 Probability and Counting

Developing Your Counting Muscles

Copyright © Cengage Learning. All rights reserved.

Conditional Probability

Probability Imagine tossing two coins and observing whether 0, 1, or 2 heads are obtained. It would be natural to guess that each of these events occurs.

Dr J Frost GCSE Sets Dr J Frost Last modified: 18th.

Final Grade Averages Weighted Averages.

Jeopardy Review Q Theoretical Probability

6.2 Find Probability Using Permutations

Math Toolkit ACCURACY, PRECISION & ERROR.

The sum of any two even integers is even.

COUNTING AND PROBABILITY

Tests of Divisibility 1 - All integers can be divided by 1

Pearson Unit 6 Topic 15: Probability 15-3: Permutations and Combinations Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007.

7.3 Sample Means HW: p. 454 (49-63 odd, 65-68).

Presentation transcript:

Probability and Counting Exercises How many different 1-operand logic operator can we have? E.g., ~ is a 1-operand logic operator. How many different 2-operand logic operator can we have? E.g., ∧, ∨, → are 2-operand logic operators.

Probability and Counting Exercises Suppose X is a finite set of x elements and Y is a finite set of y elements. How many different functions from X to Y ? How many different relations from X to Y ?

Probability and Counting Exercises In the World Cup soccer game finals, there are 32 teams that need to be divided to 8 different groups with 4 teams per group. How many different group results? If the group names (Group A,B,C,…H) matter If the group names do not matter

Probability and Counting Exercises What is the probability that a randomly chosen 4-digit integer has distinct digits and is odd?

Probability and Counting Exercises There are 100 students and three courses, CS, MATH, and ENGLISH to take. We know 28 students are taking CS, 26 students are taking MATH, and 14 are taking ENGLISH, 8 students are taking both CS and MATH, 4 students are taking both CS and ENGLISH, and 3 students are taking both MATH and ENGLISH, 2 students are taking all three courses. How many students are taking at least one course? How many students are not taking any course? How many students are taking CS and MATH but not ENGLISH? How many student are taking MATH but neither of the other two?

Probability and Counting Exercises There are two exam for a certain course. 25% students in class got 90+ in the first exam, and 15% students in class got 90+ in both exams. What percentage of students who got 90+ in the first exam also got 90+ in the second exam?

Probability and Counting Exercises Two different factories both produce a certain automobile part. The defective rate of the first factory is 2%, and the defective rate of the second factory is 5%. In a supply of 180 such automobile parts, 100 were from the first factory and 80 were from the second factory. What the probability that a part chosen at random from the 180 is from the first factory? What the probability that a part chosen at random from the 180 is defective? If a randomly chosen part is defective, what is the probability that it came from the first factory?