cKendra Kilmer February 11, 2008Section 7.1 - Experiments, Sample Spaces, and EventsDefinition: An experiment is an activity with an observable result.Definition: The outcome is the result of an experiment.Definition: The sample space is the set of all outcomes of an experiment.Definition: An event is a subset of the sample space. (Note: An event E is said to occur whenever E containsthe observed outcome.)Definition: E and F are mutually exclusive if E ∩ F = /0.Note: All of the set operations (union, intersection, complement) work the same with events.Example 1: Let’s consider the experiment of rolling a fair six-sided die and observing the number that landsuppermost.a) Determine the sample space.b) Find the event E where E = {x|x is an even number}.c) Find the event F where F = {x|x is a number greater than 2}d) Find the event (E ∩ F)e) Find the event (E ∪ F)f) Are the events E and F mutually exclusive?g) Are the events E and F complementary?Example 2: Let’s consider the experiment of flipping a fair coin two times and observing the resulting sequenceof ”heads” and ”tails”.a) Determine the sample space.b) Find the event E where E = {x|x has one or more heads}c) Find the event F where F = {x|x has more than 2 heads}d) List all events of this experiment.1cKendra Kilmer February 11, 2008Example 3: An experiment consists of casting a pair of dice and observing the number that falls uppermost oneach die.S =(1,1) (2,1) (3,1) (4,1) (5,1) (6,1)(1,2) (2,2) (3,2) (4,2) (5,2) (6,2)(1,3) (2,3) (3,3) (4,3) (5,3) (6,3)(1,4) (2,4) (3,4) (4,4) (5,4) (6,4)(1,5) (2,5) (3,5) (4,5) (5,5) (6,5)(1,6) (2,6) (3,6) (4,6) (5,6) (6,6)a) Determine the event that the sum of the numbers falling uppermost is less than or equal to 6.b) Determine the event that the number falling uppermost on one die is a 4 and the number falling uppermost onthe other die is greater than 4.Example 4: The manager of a local bank observes how long it takes a customer to complete his transactions atthe ATM.a) Describe an appropriate sample space for this experiment.b) Describe the event that it takes a customer between 2 and 3 minutes, inclusive, to complete his transactions atthe ATM.2cKendra Kilmer February 11, 2008Section 7.2 - Definition of ProbabilityDefinition: Suppose we repeat an experiment n times and an event E occurs m of those times. Thenmnis calledthe relative frequency of the event E.Example 1: Let’s say you flip a coin 100 times and a head occurs 61 times. What is the relative frequency of theevent E = {x|x is heads}?Definition: Often, the more we repeat an experiment, the more the relative frequency approaches a certain value.We call this the empirical probability of the event.Definition: The probability of an event is a number between 0 and 1 that represents the likelihood of the eventoccuring. The larger the probability, the more likely the event is to occur.Definition: An event which consists of exactly one outcome is called a simple event of the experiment.Example 2: List the simple events associated with each of the given experiments:a) A nickel and a dime are tossed, and the result of heads or tails is recorded for each coin.b) As part of a quality-control procedure, eight circuit boards are checked, and the number of defectives isrecorded.Definition: The table that lists the probability of each simple event in an experiment is known as the probabilitydistribution.Example 3: Metro Telephone Company compiled the accompanying information during a service-utilizationpertaining to the number of customers using their Dial-the-Time service from 7 A.M. to 9 A.M. on a certainweekday morning. Using these data, find the probability distribution associated with the experiment.Calls Received per Minute 10 11 12 13 14 15 16Frequency of Occurrence 6 15 30 3 30 36 03cKendra Kilmer February 11, 2008Definition: The function which assigns a probability to each of the simple events is called aprobability function. It has the following properties:1. 0 ≤ P(si) ≤ 12. P(s1) + P(s2) + ··· + P(sn) = 13. P({si} ∪ {sj}) = P(si) + P(sj) for i 6= jDefinition: Sample spaces in which the outcomes are equally likely are called uniform sample spaces. For auniform sample space S = {s1,s2,...,sn}, we can assign to the simple events s1,s2,...,snthe probabilities:P(s1) = P(s2) = ··· = P(sn) =1nFinding the probability of an event E1. Determine a sample space S associated with the experiment.2. Assign probabilities to the simple events of S.3. If E = {s1,...,sm} where {s1},...{sm} are simple events thenP(E) = P(s1) + P(s2) + ··· + P(sm)4. If E is the empty set,/0, then P(E) = 0Example 4: Let S = {s1,s2,s3,s4,s5} be the sample space associated with an experiment having the followingprobability distribution:Outcome s1s2s3s4s5Probability114314614214214Find the probability of the event:a) A = {s1,s2,s4}b) B = {s1,s5}c) C = SExample 5: If a marble is selected at random from a bowl containing three red, two white, and five black, whatis the probability that the marble drawn is not white?4cKendra Kilmer February 11, 2008Section 7.3 - Rules of ProbabilityProperties:1. P(E) ≥ 0 for any event E2. P(S) = 13. If E and F are mutually exclusive (that is, E ∩ F = /0), thenP(E ∪ F) = P(E) + P(F)4. P(E ∪F) = P(E) + P(F) − P(E ∩ F)5. P(Ec) = 1 − P(E)Example 1: Let E and F be two events of an experiment with sample space S. Suppose P(E) = 0.7, P(F) = 0.5,and P(E ∩ F) = 0.3. Compute:a) P(E ∪F)b) P(Ec)c) P(Fc)d) P(Ec∩ Fc)e) P(Ec∩ F)Example 2: An experiment consists of selecting a card at random from a 52-card deck. What is the probabilitythat a diamond or a king is drawn?5cKendra Kilmer February 11, 2008Example 3: A pair of fair six-sided dice is cast and the number that appears uppermost on each die is observed.Determine the probability of the following events:a) A double is thrown.b) The sum of the numbers is at least 4.c) At least one of the die is showing a 6.Example 4: Among 500 freshman pursuing a business degree at a university, 320 are enrolled in an Economicscourse, 225 are enrolled in a Mathematics course, and 140 are enrolled in both an Economics and a Mathematicscourse. What is the probability that a freshman selected at random from this group is enrolled in:a) an Economics or Mathematics course?b) exactly one of these two courses?c) neither an Economics course nor a Mathematics course?6cKendra Kilmer