Math 166, Spring 2012,cBenjamin Aurispa1.3 Sample Spaces and EventsAn experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, pickingmarbles out of a jar, etc. The result of an experiment is called an outcome of the experiment.The sample space of an experiment is the set of all possible outcomes. It is important to keep in mindwhat is being observed or recorded in the experiment.Example: Determine the sample space, S, for the following experiments.• Flipping a coin and observing whether it lands heads or tails.• Rolling a fair die and observing the number that is rolled.• Rolling two fair dice and observing the sum of the numbers rolled.An event is a subset of the sample space of an experiment. An elementary (or simple) event is an eventthat consists of a single outcome.Example: Consider the experiment of rolling two fair dice and observing the numbers that are rolled oneach die.The sample space S for this experiment is S =(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)The first coordinate of these ordered pairs represents the first die and the second coordinate represents thesecond die. (3, 5) and (5, 3) are different outcomes because in (3, 5), the first die rolls a 3, but in (5, 3) thesecond die rolls a 3. If it helps, think of one die as “red” and the other as “green.”• Determine the event E that the sum of the two dice is 5.• Determine the event F that the number on the first die is exactly 1 more than the number on thesecond die.• Determine the event G that a 6 is rolled.• Determine the event H that the sum of the two dice is 12.• Determine the event K that a 7 is rolled.• How many events are there total?How many simple events are there?1Math 166, Spring 2012,cBenjamin AurispaThe empty set ∅ is called the impossible event.The sample space S is called the certain event, since whatever outcome occurs is guaranteed to be in S.We can have unions, intersections, and complements of events just as before. If E and F are two events ofan experiment, then:• E ∪ F is the set of outcomes that are in E or F , i.e. E ∪ F is the event that E OR F (or both) occurs.• Ecis the set of outcomes that are not in E, i.e. Ecis the event that E does NOT occur.• E ∩ F is the set of outcomes that are in both E and F , i.e. E ∩ F is the even that both E AND F occur.If two events CANNOT happen at the same time, then E ∩F = ∅, and these events are said to be mutuallyexclusive. (The sets E and F are disjoint.)From the dice example above we saw that E = {(4, 1), (3, 2), (2, 3), (1, 4)}, F = {(2, 1), (3, 2), (4, 3), (5, 4), (6, 5)},and G = {(1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}.Are E and F mutually exclusive? Are E and G mutually exclusive?Often, tree diagrams can be used to help find the sample space.Example: Suppose I flip a coin twice and record the side that lands up on each toss.• Determine the sample space for this experiment.• Determine the event E that at least 1 tail is tossed.• Determine the event F that exactly 1 head is tossed.• Are E and F mutually exclusive?2Math 166, Spring 2012,cBenjamin AurispaA note on decks of cards: A deck of cards consists of 52 cards. There are 13 cards for each of the four suits:clubs, spades, diamonds, and hearts. The 13 cards are numbered 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King,Ace. Clubs and spades are black. Diamonds and hearts are red. A face card is a Jack, Queen, or King. (AnAce is NOT considered a face card.)Example: A fair 5-sided die is rolled, observing the number rolled, and then a card is selected from astandard deck, observing the color of the card.• Determine the sample space for this experiment.• Determine the event E that a 4 is rolled or a black card is selected.Example: A letter is selected at random from the word MATH, observing if it is a vowel or not, and then acard is randomly selected from a standard deck, observing the suit of the card. What is the sample spacefor this experiment?3Math 166, Spring 2012,cBenjamin Aurispa1.4 Basics of ProbabilityDefinition: A sample space S in which all outcomes are equally likely is called a uniform sample space.If S is a finite uniform sample space and E is any event, then the probability of E, P (E), is given by:P (E) =Number of ways for E to occurTotal number of possible outcomes in S=n(E)n(S)Note: Probabilities will ALWAYS be between 0 and 1, inclusive. The larger the probability, the more likelyit is to occur.Example: Suppose a fair die is rolled and the number that lands up is recorded. The sample space for thisexperiment is S = {1, 2, 3, 4, 5, 6}.• Is this a uniform sample space?• What is the probability that an even number is rolled?• What is the probability that a number less than 3 is rolled?Example: A card is drawn from a standard deck of cards. What is the probability that:a Jack is drawn? A club? A face card?Example: Consider the experiment of rolling two fair dice and observing the numbers that land up. Wealready found the sample space.• What is the probability that the sum of the numbers on the two dice is 9?• What is the probability that the sum of the dice is more than 10?• What is the probability that a 5 is rolled?• What is the probability that a double is not rolled?4Math 166, Spring 2012,cBenjamin AurispaExample: Consider the composition of a three-child family in which the children were born at differenttimes. Assume that a girl is as likely as a boy at each birth.• What is the sample space for this “experiment?”• What is the probability that there is exactly 1 boy in the family?• What is the probability that there are at least two boys in the family?Sometimes experiments are run to help estimate the probability of certain events. Probabilities that arebased on collected data are called empirical probabilities.If an experiment is performed n times and an event E occurs m times, then the relative frequency of theevent E ismn.Example: In a survey conducted to see how long Americans keep their cars, a group of 2000 car ownerswere asked how long they plan to keep their