More Probability: Poker Hands and some issues in Counting


 Dylan Rodgers
 4 years ago
 Views:
Transcription
1 More Probability: Poker Hands and some issues in Counting
2 Data From Thursday Everybody flipped a pair of coins and recorded how many times they got two heads, two tails, or one of each. We saw that the probability of getting two heads is.25 and that the probability of getting one of each is.50. The summary of the data is as follows. 2
3 Outcome Two Heads Two Tails One of Each Number Fraction
4 Everybody also rolled two dice 20 times and determined the sum. The tabulation of the data is as follows. 4
5 Sum of Two Dice Sum Number Fraction
6 The theoretical probability of getting a sum of 12 (e.g., rolling a double six) is 1/36, since of the 36 outcomes of rolling two dice, one of them results in a sum of 12. In decimal form, this is about.028, or very close to what the class data was. 6
7 Poker Hands 7
8 Poker is played with a deck of 52 cards. Each card has a suit and a value. 8
9 The suits are spades hearts diamonds and clubs. 9
10 There are 13 values: 2 through 10, Jack, Queen, King, and Ace. A poker hand is made up of 5 cards. The different poker hands are, from best to worst, are: 10
11 Royal Flush Straight Flush Four of a Kind 11
12 Full House Flush Straight 12
13 3 of a kind 2 pair 1 pair 13
14 Hand Example Royal Flush Straight Flush A, K, Q, J, 10 J, 10, 9, 8, 7 4 of a kind 8, 8, 8, 8, J Full House Flush Straight 4, 4, 4, J, J K, 10, 8, 7, 3 Q, J, 10, 9, 8 3 of a kind 8, 8, 8, J, 6 2 pair 8, 8, J, J, A 1 pair 8, 8, 10, J, K 14
15 Royal flush: A, K, Q, J, 10 in the same suit. Straight flush: 5 consecutive cards in the same suit. 4 of a kind: 4 cards of the same value. Full house: 3 cards of the same value and a pair. 15
16 Flush: 5 cards in the same suit. Straight: 5 consecutive cards (of any suit). An ace can be low or high. 3 of a kind: 3 cards of the same value. 1 pair: 2 cards of the same value. 16
17 How many poker hands are there? The number is how many ways you can choose 5 cards out of 52. This number is usually denoted 52C5, or ( 52 52C5, or 5 ). It is also called a binomial coefficient. 17
18 How many ways are there to choose 1 item out of 5? There are 5 ways. How many ways are there to choose 2 items out of 4? If the items are labeled a, b, c, d, we could have {a,b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, so there are 6 ways, provided that the order in which we choose them does not matter (like in a poker hand). 18
19 There is a formula to compute binomial coefficients for any pair of numbers. It is n! ncr = = r! * (nr)! n*(n1)*... *(nr+1) r*(r1)*... *2*1 where n! means 1 * 2 * 3 *... * n 19
20 Many scientific calculators have n! buttons, and many have buttons, or menu items to calculate ncr. Some values of n! are: 0! = 1 1! = 1 2! = 2 3! = 6 4! = 24 20
21 As an example of this formula, we see that nc1 = n for any number. What this means is that if you choose 1 item out of n, there are n ways to do this. Another example is that nc2 = n*(n1)/2. This comes from canceling terms in the formula. 21
22 For example, 4C2 = 4! (2! * 2!) = 24 (2*2) = 6 Alternatively, 4C2 = 4*3/2 = 6. 22
23 If you calculate 52C5, you will get 52C5 = 2,598,960. Thus, there are 2,598,960 possible 5 card poker hands. Excel can compute ncr and n! It uses the commands =combin(n,r) and =fact(n), respectively. 23
24 What is the probability of getting a royal flush when dealt 5 cards? There are 4 ways to get a royal flush, since the only choice is which suit you get. So, the probability is which is about 1 out of 600,000 deals. 24
25 In order to compute the probability of other hands, one approach is to decide how many things you need to choose in order to write down a hand, and then determine how many ways each of the choices can occur. For example, to count royal flushes, we saw that you only have to choose a suit. 25
26 Counting Independent Events If one event does not affect the outcome of another, they are called independent. To count the number of ways a pair of independent events can occur, multiply the number of ways each way can occur. 26
27 Example Rolling two dice is an example of two independent events: what you get on one die does not affect what can happen on the other. Since there are 6 outcomes for rolling one die, there are 6 * 6 = 36 outcomes for rolling two dice. 27
28 How many ways are there to get a straight flush? To choose a given straight flush, you must choose a suit, and a starting (or ending) value for the 5 in a row. There are 4 choices for the suit. What suit you choose does not affect the choice of starting value for the 5 in a row. 28
29 There are 10 possible starting values (A through 10). However, if we want a straight flush which is not a royal flush, we cannot start at 10, so there are 9 choices. Therefore, there are 4 * 9 = 36 total straight flushes. 29
30 The probability of a straight flush (which is not a royal flush) is then
31 What is the probability of a 4 of a kind? An example is 8, 8, 8, 8, J. We must count how many 4 of a kinds there are. To get a 4 of a kind, you must choose the value of the 4 of a kind, and choose the remaining card. There are 13 choices for the value of the 4 of a kind. The 5th card can be any of the remaining 48 cards. So there are 48 choices for it. 31
32 The number of four of a kinds is then 13 * 48 = 624. and so the probability of a full house is
33 A full house consists of a 3 of a kind and a 2 of a kind. What is the probability of getting a full house? An example is 4, 4, 4, J, J. To have a full house you must choose the value of a 3 of a kind and the value of a 2 of a kind. You must also choose which 3 cards make up the 3 of a kind and which 2 make up the 2 of a kind. This is one of the more complicated counts. 33
34 There are 13C1 = 13 ways to choose the value of the 3 of a kind. There are then 12C1 = 12 ways to choose the value of the pair. There are 4C3 = 4 ways to choose the three cards for the 3 of a kind. There are 4C2 = 6 ways to choose the 2 cards for the pair. 34
35 So, the number of ways to get a full house is 13 * 12 * 4 * 6 = The probability of a full house is then a little worse than 1 out of 4000 hands. 35
Poker: Probabilities of the Various Hands
Poker: Probabilities of the Various Hands 22 February 2012 Poker II 22 February 2012 1/27 Some Review from Monday There are 4 suits and 13 values. The suits are Spades Hearts Diamonds Clubs There are 13
More informationPoker: Probabilities of the Various Hands
Poker: Probabilities of the Various Hands 19 February 2014 Poker II 19 February 2014 1/27 Some Review from Monday There are 4 suits and 13 values. The suits are Spades Hearts Diamonds Clubs There are 13
More informationPoker: Further Issues in Probability. Poker I 1/29
Poker: Further Issues in Probability Poker I 1/29 How to Succeed at Poker (3 easy steps) 1 Learn how to calculate complex probabilities and/or memorize lots and lots of pokerrelated probabilities. 2 Take
More informationPoker Hands. Christopher Hayes
Poker Hands Christopher Hayes Poker Hands The normal playing card deck of 52 cards is called the French deck. The French deck actually came from Egypt in the 1300 s and was already present in the Middle
More informationClassical vs. Empirical Probability Activity
Name: Date: Hour : Classical vs. Empirical Probability Activity (100 Formative Points) For this activity, you will be taking part in 5 different probability experiments: Rolling dice, drawing cards, drawing
More informationThe point value of each problem is in the lefthand margin. You must show your work to receive any credit, except on problems 1 & 2. Work neatly.
Introduction to Statistics Math 1040 Sample Exam II Chapters 57 4 Problem Pages 4 Formula/Table Pages Time Limit: 90 Minutes 1 No Scratch Paper Calculator Allowed: Scientific Name: The point value of
More informationChapter 2. Permutations and Combinations
2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find
More informationUnit 6: Probability. Marius Ionescu 10/06/2011. Marius Ionescu () Unit 6: Probability 10/06/ / 22
Unit 6: Probability Marius Ionescu 10/06/2011 Marius Ionescu () Unit 6: Probability 10/06/2011 1 / 22 Chapter 13: What is a probability Denition The probability that an event happens is the percentage
More information{ a, b }, { a, c }, { b, c }
12 d.) 0(5.5) c.) 0(5,0) h.) 0(7,1) a.) 0(6,3) 3.) Simplify the following combinations. PROBLEMS: C(n,k)= the number of combinations of n distinct objects taken k at a time is COMBINATION RULE It can easily
More informationCounting Poker Hands
Counting Poker Hands George Ballinger In a standard deck of cards there are kinds of cards: ce (),,,,,,,,,, ack (), ueen () and ing (). Each of these kinds comes in four suits: Spade (), Heart (), Diamond
More informationDef: The intersection of A and B is the set of all elements common to both set A and set B
Def: Sample Space the set of all possible outcomes Def: Element an item in the set Ex: The number "3" is an element of the "rolling a die" sample space Main concept write in Interactive Notebook Intersection:
More informationUnit 6: Probability. Marius Ionescu 10/06/2011. Marius Ionescu () Unit 6: Probability 10/06/ / 22
Unit 6: Probability Marius Ionescu 10/06/2011 Marius Ionescu () Unit 6: Probability 10/06/2011 1 / 22 Chapter 13: What is a probability Denition The probability that an event happens is the percentage
More informationIf you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics
If you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics probability that you get neither? Class Notes The Addition Rule (for OR events) and Complements
More information2.5 Sample Spaces Having Equally Likely Outcomes
Sample Spaces Having Equally Likely Outcomes 3 Sample Spaces Having Equally Likely Outcomes Recall that we had a simple example (fair dice) before on equallylikely sample spaces Since they will appear
More informationNovember 6, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern
More informationUNIT 4 APPLICATIONS OF PROBABILITY Lesson 1: Events. Instruction. Guided Practice Example 1
Guided Practice Example 1 Bobbi tosses a coin 3 times. What is the probability that she gets exactly 2 heads? Write your answer as a fraction, as a decimal, and as a percent. Sample space = {HHH, HHT,
More informationKey Concepts. Theoretical Probability. Terminology. Lesson 111
Key Concepts Theoretical Probability Lesson  Objective Teach students the terminology used in probability theory, and how to make calculations pertaining to experiments where all outcomes are equally
More informationProbability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College
Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical
More information4.1 Sample Spaces and Events
4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an
More informationMath 166: Topics in Contemporary Mathematics II
Math 166: Topics in Contemporary Mathematics II Xin Ma Texas A&M University September 30, 2017 Xin Ma (TAMU) Math 166 September 30, 2017 1 / 11 Last Time Factorials For any natural number n, we define
More informationGAMBLING ( ) Name: Partners: everyone else in the class
Name: Partners: everyone else in the class GAMBLING Games of chance, such as those using dice and cards, oporate according to the laws of statistics: the most probable roll is the one to bet on, and the
More informationProbability is often written as a simplified fraction, but it can also be written as a decimal or percent.
CHAPTER 1: PROBABILITY 1. Introduction to Probability L EARNING TARGET: I CAN DETERMINE THE PROBABILITY OF AN EVENT. What s the probability of flipping heads on a coin? Theoretically, it is 1/2 1 way to
More information3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1 Suppose P (A 0, P (B 05 (a If A and B are independent, what is P (A B? What is P (A B? (b If A and B are disjoint, what is
More informationn(s)=the number of ways an event can occur, assuming all ways are equally likely to occur. p(e) = n(e) n(s)
The following story, taken from the book by Polya, Patterns of Plausible Inference, Vol. II, Princeton Univ. Press, 1954, p.101, is also quoted in the book by Szekely, Classical paradoxes of probability
More informationCompound Probability. Set Theory. Basic Definitions
Compound Probability Set Theory A probability measure P is a function that maps subsets of the state space Ω to numbers in the interval [0, 1]. In order to study these functions, we need to know some basic
More information4.3 Rules of Probability
4.3 Rules of Probability If a probability distribution is not uniform, to find the probability of a given event, add up the probabilities of all the individual outcomes that make up the event. Example:
More informationIntermediate Math Circles November 1, 2017 Probability I
Intermediate Math Circles November 1, 2017 Probability I Probability is the study of uncertain events or outcomes. Games of chance that involve rolling dice or dealing cards are one obvious area of application.
More informationBasic Concepts of Probability and Counting Section 3.1
Basic Concepts of Probability and Counting Section 3.1 Summer 2013  Math 1040 June 17 (1040) M 10403.1 June 17 1 / 12 Roadmap Basic Concepts of Probability and Counting Pages 128137 Counting events,
More informationReview. Natural Numbers: Whole Numbers: Integers: Rational Numbers: Outline Sec Comparing Rational Numbers
FOUNDATIONS Outline Sec. 31 Gallo Name: Date: Review Natural Numbers: Whole Numbers: Integers: Rational Numbers: Comparing Rational Numbers Fractions: A way of representing a division of a whole into
More informationSimple Probability. Arthur White. 28th September 2016
Simple Probability Arthur White 28th September 2016 Probabilities are a mathematical way to describe an uncertain outcome. For eample, suppose a physicist disintegrates 10,000 atoms of an element A, and
More informationProbability Simulation User s Manual
Probability Simulation User s Manual Documentation of features and usage for Probability Simulation Copyright 2000 Corey Taylor and Rusty Wagner 1 Table of Contents 1. General Setup 3 2. Coin Section 4
More informationDefine and Diagram Outcomes (Subsets) of the Sample Space (Universal Set)
12.3 and 12.4 Notes Geometry 1 Diagramming the Sample Space using Venn Diagrams A sample space represents all things that could occur for a given event. In set theory language this would be known as the
More informationProbability Homework Pack 1
Dice 2 Probability Homework Pack 1 Probability Investigation: SKUNK In the game of SKUNK, we will roll 2 regular 6sided dice. Players receive an amount of points equal to the total of the two dice, unless
More informationProbability Theory. Mohamed I. Riffi. Islamic University of Gaza
Probability Theory Mohamed I. Riffi Islamic University of Gaza Table of contents 1. Chapter 1 Probability Properties of probability Counting techniques 1 Chapter 1 Probability Probability Theorem P(φ)
More informationMultiplication and Probability
Problem Solving: Multiplication and Probability Problem Solving: Multiplication and Probability What is an efficient way to figure out probability? In the last lesson, we used a table to show the probability
More informationSection 7.1 Experiments, Sample Spaces, and Events
Section 7.1 Experiments, Sample Spaces, and Events Experiments An experiment is an activity with observable results. 1. Which of the follow are experiments? (a) Going into a room and turning on a light.
More informationDiscrete Random Variables Day 1
Discrete Random Variables Day 1 What is a Random Variable? Every probability problem is equivalent to drawing something from a bag (perhaps more than once) Like Flipping a coin 3 times is equivalent to
More informationChapter 1: Sets and Probability
Chapter 1: Sets and Probability Section 1.31.5 Recap: Sample Spaces and Events An is an activity that has observable results. An is the result of an experiment. Example 1 Examples of experiments: Flipping
More informationHere are two situations involving chance:
Obstacle Courses 1. Introduction. Here are two situations involving chance: (i) Someone rolls a die three times. (People usually roll dice in pairs, so dice is more common than die, the singular form.)
More informationCISC102 Fall 2017 Week 8
Week 8 Page! of! 34 Playing cards. CISC02 Fall 207 Week 8 Some of the following examples make use of the standard 52 deck of playing cards as shown below. There are 4 suits (clubs, spades, hearts, diamonds)
More informationProbability Review 41
Probability Review 41 For the following problems, give the probability to four decimals, or give a fraction, or if necessary, use scientific notation. Use P(A) = 1  P(not A) 1) A coin is tossed 6 times.
More informationDeveloped by Rashmi Kathuria. She can be reached at
Developed by Rashmi Kathuria. She can be reached at . Photocopiable Activity 1: Step by step Topic Nature of task Content coverage Learning objectives Task Duration Arithmetic
More informationUp & Down GOAL OF THE GAME UP&DOWN CARD A GAME BY JENS MERKL & JEANCLAUDE PELLIN ART BY CAMILLE CHAUSSY
Up & Down A GAME BY JENS MERKL & JEANCLAUDE PELLIN ART BY CAMILLE CHAUSSY GOAL OF THE GAME UP&DOWN is a trick taking game with plenty of ups and downs. This is because prior to each trick, one of the
More information7.1 Experiments, Sample Spaces, and Events
7.1 Experiments, Sample Spaces, and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment
More informationContemporary Mathematics Math 1030 Sample Exam I Chapters Time Limit: 90 Minutes No Scratch Paper Calculator Allowed: Scientific
Contemporary Mathematics Math 1030 Sample Exam I Chapters 1315 Time Limit: 90 Minutes No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the lefthand margin.
More informationDate. Probability. Chapter
Date Probability Contests, lotteries, and games offer the chance to win just about anything. You can win a cup of coffee. Even better, you can win cars, houses, vacations, or millions of dollars. Games
More information6/24/14. The Poker Manipulation. The Counting Principle. MAFS.912.SIC.1: Understand and evaluate random processes underlying statistical experiments
The Poker Manipulation Unit 5 Probability 6/24/14 Algebra 1 Ins1tute 1 6/24/14 Algebra 1 Ins1tute 2 MAFS. 7.SP.3: Investigate chance processes and develop, use, and evaluate probability models MAFS. 7.SP.3:
More informationChapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance FreeResponse 1. A spinner with regions numbered 1 to 4 is spun and a coin is tossed. Both the number spun and whether the coin lands heads or tails is
More informationABE/ASE Standards Mathematics
[Lesson Title] TEACHER NAME PROGRAM NAME Program Information Playing the Odds [Unit Title] Data Analysis and Probability NRS EFL(s) 3 4 TIME FRAME 240 minutes (double lesson) ABE/ASE Standards Mathematics
More informationSTAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes
STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes Pengyuan (Penelope) Wang May 25, 2011 Review We have discussed counting techniques in Chapter 1. (Principle
More informationPermutations: The number of arrangements of n objects taken r at a time is. P (n, r) = n (n 1) (n r + 1) =
Section 6.6: Mixed Counting Problems We have studied a number of counting principles and techniques since the beginning of the course and when we tackle a counting problem, we may have to use one or a
More information1. How to identify the sample space of a probability experiment and how to identify simple events
Statistics Chapter 3 Name: 3.1 Basic Concepts of Probability Learning objectives: 1. How to identify the sample space of a probability experiment and how to identify simple events 2. How to use the Fundamental
More informationGrade 8 Math Assignment: Probability
Grade 8 Math Assignment: Probability Part 1: Rock, Paper, Scissors  The Study of Chance Purpose An introduction of the basic information on probability and statistics Materials: Two sets of hands Paper
More informationChapter 4: Introduction to Probability
MTH 243 Chapter 4: Introduction to Probability Suppose that we found that one of our pieces of data was unusual. For example suppose our pack of M&M s only had 30 and that was 3.1 standard deviations below
More informationA Probability Work Sheet
A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair sixsided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we
More informationUnit 9: Probability Assignments
Unit 9: Probability Assignments #1: Basic Probability In each of exercises 1 & 2, find the probability that the spinner shown would land on (a) red, (b) yellow, (c) blue. 1. 2. Y B B Y B R Y Y B R 3. Suppose
More informationDISCUSSION #8 FRIDAY MAY 25 TH Sophie Engle (Teacher Assistant) ECS20: Discrete Mathematics
DISCUSSION #8 FRIDAY MAY 25 TH 2007 Sophie Engle (Teacher Assistant) ECS20: Discrete Mathematics 2 Homework 8 Hints and Examples 3 Section 5.4 Binomial Coefficients Binomial Theorem 4 Example: j j n n
More information3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,
More informationWhen a number cube is rolled once, the possible numbers that could show face up are
C3 Chapter 12 Understanding Probability Essential question: How can you describe the likelihood of an event? Example 1 Likelihood of an Event When a number cube is rolled once, the possible numbers that
More informationBasic Probability Models. PingShou Zhong
asic Probability Models PingShou Zhong 1 Deterministic model n experiment that results in the same outcome for a given set of conditions Examples: law of gravity 2 Probabilistic model The outcome of the
More informationConditional Probability Worksheet
Conditional Probability Worksheet P( A and B) P(A B) = P( B) Exercises 36, compute the conditional probabilities P( AB) and P( B A ) 3. P A = 0.7, P B = 0.4, P A B = 0.25 4. P A = 0.45, P B = 0.8, P A
More informationNovember 8, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 8, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Crystallographic notation The first symbol
More informationProbability: introduction
May 6, 2009 Probability: introduction page 1 Probability: introduction Probability is the part of mathematics that deals with the chance or the likelihood that things will happen The probability of an
More informationMixed Counting Problems
We have studied a number of counting principles and techniques since the beginning of the course and when we tackle a counting problem, we may have to use one or a combination of these principles. The
More informationProbability. The Bag Model
Probability The Bag Model Imagine a bag (or box) containing balls of various kinds having various colors for example. Assume that a certain fraction p of these balls are of type A. This means N = total
More informationMC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES
MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES Thursday, 4/17/14 The Addition Principle The InclusionExclusion Principle The Pigeonhole Principle Reading: [J] 6.1, 6.8 [H] 3.5, 12.3 Exercises:
More informationActivity 1: Play comparison games involving fractions, decimals and/or integers.
Students will be able to: Lesson Fractions, Decimals, Percents and Integers. Play comparison games involving fractions, decimals and/or integers,. Complete percent increase and decrease problems, and.
More informationImportant Distributions 7/17/2006
Important Distributions 7/17/2006 Discrete Uniform Distribution All outcomes of an experiment are equally likely. If X is a random variable which represents the outcome of an experiment of this type, then
More informationIndependence Is The Word
Problem 1 Simulating Independent Events Describe two different events that are independent. Describe two different events that are not independent. The probability of obtaining a tail with a coin toss
More informationMath 227 Elementary Statistics. Bluman 5 th edition
Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 4 Probability and Counting Rules 2 Objectives Determine sample spaces and find the probability of an event using classical probability or empirical
More informationChapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
More informationI. WHAT IS PROBABILITY?
C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and
More informationSection 6.1 #16. Question: What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit?
Section 6.1 #16 What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1
More informationMost of the time we deal with theoretical probability. Experimental probability uses actual data that has been collected.
AFM Unit 7 Day 3 Notes Theoretical vs. Experimental Probability Name Date Definitions: Experiment: process that gives a definite result Outcomes: results Sample space: set of all possible outcomes Event:
More informationDiscrete Structures Lecture Permutations and Combinations
Introduction Good morning. Many counting problems can be solved by finding the number of ways to arrange a specified number of distinct elements of a set of a particular size, where the order of these
More informationMath 14 Lecture Notes Ch. 3.3
3.3 Two Basic Rules of Probability If we want to know the probability of drawing a 2 on the first card and a 3 on the 2 nd card from a standard 52card deck, the diagram would be very large and tedious
More informationPart 1: I can express probability as a fraction, decimal, and percent
Name: Pattern: Part 1: I can express probability as a fraction, decimal, and percent For #1 to #4, state the probability of each outcome. Write each answer as a) a fraction b) a decimal c) a percent Example:
More informationIf a regular sixsided die is rolled, the possible outcomes can be listed as {1, 2, 3, 4, 5, 6} there are 6 outcomes.
Section 11.1: The Counting Principle 1. Combinatorics is the study of counting the different outcomes of some task. For example If a coin is flipped, the side facing upward will be a head or a tail the
More informationJIGSAW ACTIVITY, TASK # Make sure your answer in written in the correct order. Highest powers of x should come first, down to the lowest powers.
JIGSAW ACTIVITY, TASK #1 Your job is to multiply and find all the terms in ( 1) Recall that this means ( + 1)( + 1)( + 1)( + 1) Start by multiplying: ( + 1)( + 1) x x x x. x. + 4 x x. Write your answer
More informationConditional Probability Worksheet
Conditional Probability Worksheet EXAMPLE 4. Drug Testing and Conditional Probability Suppose that a company claims it has a test that is 95% effective in determining whether an athlete is using a steroid.
More informationCounting Methods and Probability
CHAPTER Counting Methods and Probability Many good basketball players can make 90% of their free throws. However, the likelihood of a player making several free throws in a row will be less than 90%. You
More informationINDIAN STATISTICAL INSTITUTE
INDIAN STATISTICAL INSTITUTE B1/BVR Probability Home Assignment 1 200707 1. A poker hand means a set of five cards selected at random from usual deck of playing cards. (a) Find the probability that it
More informationNovember 11, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 11, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Probability Rules Probability Rules Rule 1.
More informationStatistics 1040 Summer 2009 Exam III
Statistics 1040 Summer 2009 Exam III 1. For the following basic probability questions. Give the RULE used in the appropriate blank (BEFORE the question), for each of the following situations, using one
More information(a) Suppose you flip a coin and roll a die. Are the events obtain a head and roll a 5 dependent or independent events?
Unit 6 Probability Name: Date: Hour: Multiplication Rule of Probability By the end of this lesson, you will be able to Understand Independence Use the Multiplication Rule for independent events Independent
More informationCounting integral solutions
Thought exercise 2.2 20 Counting integral solutions Question: How many nonnegative integer solutions are there of x 1 +x 2 +x 3 +x 4 = 10? Thought exercise 2.2 20 Counting integral solutions Question:
More informationProbability, Continued
Probability, Continued 12 February 2014 Probability II 12 February 2014 1/21 Last time we conducted several probability experiments. We ll do one more before starting to look at how to compute theoretical
More informationSection 7.3 and 7.4 Probability of Independent Events
Section 7.3 and 7.4 Probability of Independent Events Grade 7 Review Two or more events are independent when one event does not affect the outcome of the other event(s). For example, flipping a coin and
More informationFundamentals of Probability
Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible
More informationc. If you roll the die six times what are your chances of getting at least one d. roll.
1. Find the area under the normal curve: a. To the right of 1.25 (10078.87)/2=10.565 b. To the left of 0.40 (10031.08)/2=34.46 c. To the left of 0.80 (10057.63)/2=21.185 d. Between 0.40 and 1.30 for
More informationPan (7:30am) Juan (8:30am) Juan (9:30am) Allison (10:30am) Allison (11:30am) Mike L. (12:30pm) Mike C. (1:30pm) Grant (2:30pm)
STAT 225 FALL 2012 EXAM ONE NAME Your Section (circle one): Pan (7:30am) Juan (8:30am) Juan (9:30am) Allison (10:30am) Allison (11:30am) Mike L. (12:30pm) Mike C. (1:30pm) Grant (2:30pm) Grant (3:30pm)
More informationINDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2
INDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2 WARM UP Students in a mathematics class pick a card from a standard deck of 52 cards, record the suit, and return the card to the deck. The results
More informationProbability. Ms. Weinstein Probability & Statistics
Probability Ms. Weinstein Probability & Statistics Definitions Sample Space The sample space, S, of a random phenomenon is the set of all possible outcomes. Event An event is a set of outcomes of a random
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More 9.9.3 Practice Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. ) In how many ways can you answer the questions on
More informationMat 344F challenge set #2 Solutions
Mat 344F challenge set #2 Solutions. Put two balls into box, one ball into box 2 and three balls into box 3. The remaining 4 balls can now be distributed in any way among the three remaining boxes. This
More informationTextbook: pp Chapter 2: Probability Concepts and Applications
1 Textbook: pp. 3980 Chapter 2: Probability Concepts and Applications 2 Learning Objectives After completing this chapter, students will be able to: Understand the basic foundations of probability analysis.
More informationNormal Distribution Lecture Notes Continued
Normal Distribution Lecture Notes Continued 1. Two Outcome Situations Situation: Two outcomes (for against; heads tails; yes no) p = percent in favor q = percent opposed Written as decimals p + q = 1 Why?
More information, the of all of a probability experiment. consists of outcomes. (b) List the elements of the event consisting of a number that is greater than 4.
41 Sample Spaces and Probability as a general concept can be defined as the chance of an event occurring. In addition to being used in games of chance, probability is used in the fields of,, and forecasting,
More informationA. 15 B. 24 C. 45 D. 54
A spinner is divided into 8 equal sections. Lara spins the spinner 120 times. It lands on purple 30 times. How many more times does Lara need to spin the spinner and have it land on purple for the relative
More informationJunior Circle Meeting 5 Probability. May 2, ii. In an actual experiment, can one get a different number of heads when flipping a coin 100 times?
Junior Circle Meeting 5 Probability May 2, 2010 1. We have a standard coin with one side that we call heads (H) and one side that we call tails (T). a. Let s say that we flip this coin 100 times. i. How
More information