CHAPTER 4 Probability and Counting Rules Student Notes Stat 200 Elementary Statistics Introduction Probability Section 4 1 Sample Spaces and Probability Probability Experiment Outcome Sample Space Event Examples 1 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 children Finding 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 Child Second Child Third Child Outcomes Event Simple Event Compound Event Equally likely events Venn Diagrams Three Interpretations of Probability Classical Probability P E n E Number of outcomes in E n S total number of oucomes in the sample space Rounding Rule Probability Rules 1 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 3 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 Examples 1 For a card drawn from an ordinary deck find the probability of getting a a queen b the 6 of clubs c a 3 or a diamond d a 3 or a 6 2 If a family has three children find the probability that all the children are girls Complement of an event Rule for complementary Events Examples Find the complement of each event 1 Rolling a die and getting a 4 2 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 weekday Empirical Probability Subjective Probability Law of Large Numbers 4 2 The Addition Rules for Probability Examples Consider 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 Exclusive Determine 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 Rule Rule 1 Rule 2 Examples 1 A day of the week is selected at random Find the probability that it is a weekend day 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 2 3 A single card is drawn from a deck Find the probability that it is a king or a club Example Titanic Men 332 Survived Died 1360 Total 1692 Women 318 104 422 Boys 29 35 18 64 56 Girls Totals 27 706 1517 2223 Find the probability of randomly selecting a man or a boy 4 3 The Multiplication Rule Independent Dependent Multiplication Rule 1 Examples 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 is drawn Find the probability of getting a queen and then an ace Conditional Probability Multiplication 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 Example Box 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 Probability Example The 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 is drawn 4 4 Counting Rule Multiplication Rule for a sequence of events Permutations 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 1 how many different bytes are possible and each bit must be a 0 or 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 1 and 51