CHAPTER 4 Probability and Counting RulesStat 200 – Elementary StatisticsIntroductionSection 4.1 Sample Spaces and ProbabilityComplement of an eventSubjective ProbabilityMutually ExclusiveExamplesFormula for Conditional ProbabilityExampleExample - Probabilities for “At Least”Stat 200 – Elementary Statistics Introduction ProbabilitySection 4.1 Sample Spaces and ProbabilityProbability ExperimentOutcomeSample SpaceEventExamples1. Find the sample space for flipping a coin and rolling a die.2. Find the sample space for drawing one card from an ordinary deck of cards.3. Find the sample space for the gender of children if a family has three childrenFinding a sample space – A tree diagram is a device consisting of line segments emanating from a starting point and also from the outcome point. It is used to determine all possible outcomes of a probability experiment.Reworking the sample space for the gender of children if a family has three children using a tree diagram.First Second ThirdChild Child Child OutcomesStudent NotesCHAPTER 4 Probability and Counting RulesEventSimple EventCompound EventEqually likely eventsVenn DiagramsThree Interpretations of ProbabilityClassical ProbabilityRounding RuleProbability Rules1. The probability of an event E is a number (either a fraction or decimal) between and including 0 and 1. This is denoted by:2. If an event E cannot occur (i.e., the event contains no members in the sample space), the probability is _______.space sample in the oucomes ofnumber totalEin outcomes ofNumber )()()( SnEnEP3. If an event E is certain, then the probability of E is ________.4. The sum of the probabilities of the outcomes in the sample space is _____.Examples1. For a card drawn from an ordinary deck, find the probability of getting a) a queenb) the 6 of clubsc) a 3 or a diamondd) a 3 or a 62. If a family has three children, find the probability that all the children are girls.Complement of an eventRule for complementary EventsExamplesFind the complement of each event.1. Rolling a die and getting a 42. Selecting a letter of the alphabet and getting a vowel.3. Selecting a month and getting a month that begins with a J.4. Selecting a day of the week and getting a weekdayEmpirical ProbabilitySubjective ProbabilityLaw of Large Numbers4.2 The Addition Rules for ProbabilityExamplesConsider the problem of selecting at random a card from a standard deck and finding the probability:a. the card is a king or is a diamond.b. the card is a king or a ten .Mutually ExclusiveDetermine which events are mutually exclusive and which are not, when a single die is rolled.a. Getting an odd number and getting an even number.b. Getting a 3 and getting an odd number.c. Getting an odd number and getting a number less than 4.d. Getting a number greater than 4 and getting a number less than 4.Addition RuleRule 1:Rule 2:Examples1. A day of the week is selected at random. Find the probability that it is a weekend day.2. A bag contains 3 red marbles, 4 yellow marbles and 5 blue marbles. If a person selects a marble at random, find the probability that it is either a red marble or a blue marble.3. A single card is drawn from a deck. Find the probability that it is a king or a club.Example: Titanic 4.3 The Multiplication RuleIndependentDependentMultiplication Rule 1:Examples Men Women Boys Girls TotalsSurvived 332 318 29 27 706Died 1360 104 35 18 1517Total 1692 422 64 56 2223Find the probability of randomly selecting a man or a boy.1. A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die.2. A card is drawn from a deck and replaced; then a second card isdrawn. Find the probability of getting a queen and then an ace. Conditional ProbabilityMultiplication Rule 2: 1. A person owns a collection of 30 CDs, of which 5 are country music. If 2 CDs are selected at random, find the probability that both are country music.2. Three cards are drawn from an ordinary deck and not replaced. Find the probability of these.a. Getting 3 jacks.b. Getting an ace, a king, and a queen in order.c. Getting a club, a spade, and a heart in order.d. Getting 3 clubs.ExampleBox 1 contains 2 red balls and 1 blue ball. Box 2 contains 3 blue balls and 1 red ball. A coin is tossed. If it falls heads up, box 1 is selected and a ball is drawn. If it falls tails up, box 2 is selected and a ball is drawn. Find the probability of selecting a red ball.Formula for Conditional ProbabilityExampleThe probability that Sam parks in a no-parking zone and gets a parking ticket is 0.06, and the probability that Sam cannot find a legal parking space and has to park in the no-parking zone is 0.20. On Tuesday, Sam arrives at school and has to park in a no-parking zone. Find the probability that he will get a parking ticket.Example - Probabilities for “At Least”A game is played by drawing four cards from an ordinary deck and replacing each card after it is drawn. Find the probability of winning if at least one ace isdrawn.4.4 Counting RuleMultiplication Rule for a sequence of eventsPermutations#1. How many possibilities are there to form a code consisting of a letter followed by a digit?#2. If a byte is defined to be a sequence of 8 bits and each bit must be a 0 or 1, how many different bytes are possible? #3. How many different ways are there to rearranging 5 questions on a survey?#4. How many different routes are there to travel between 3 cities?#5. How many different routes are there to travel the fifty states?#6. How many different routes are there to travel 4 of 50 states?#7. How many different ways can you select three songs to play out of 10, if order is important?Combinations#8. The Board of Trustees has 9 members.a. How many different 3-person committees are possible?b. When the board elects 3 officers, how many different slates of candidates are possible?#9 How many different ways are there to select 6 different numbers between 1and