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MATH 141-501 Section 7.1 Lecture Notes An experiment is an activity with observable results. The results of an experiments are called outcomes. What are some examples of experiments? Mathematical terms The sample space is the set of all outcomes of an experiment. An outcome of an experiment is an element in the sample space and is called a sample point. An event is a subset of the sample space, i.e. an event is a collection of events. We say that A occurs if A contains the observed outcome. Note that the empty set is an event (which never occurs). The sample space S is an event (which always occurs). Example: Describe the sample space associated with randomly drawing a marble from a box which contains red marbles, blue marbles, and green marbles. What are the sample points? What are the events? 1 If two events can not both occur at the same time, they are said to be mutually exclusive. From a mathematical point of view, two events A and B are mutually exclusive if A ∩ B = ∅. Describe the sample space associated to the experiment of flipping a coin four times and observing which side (heads or tails) lands up each time. What is the event, A, that exactly one toss is heads? What is the event, B, that the second and third tosses are tails? Are A and B mutually exclusive? 2 Two fair six-sided dice are rolled (one red, one blue), and the results are recorded. What is the sample space associated with this experiment? Write the event U that the sum of the two dice is 5. Write the event V that each dice shows the same number after being rolled. Are U and V mutually exclusive? 3 Example ( based on an example from Joe Kahlig’s lecture notes): A foursided die is rolled. If the result of the roll is odd, then the die is rolled a second time. The sum of the numbers rolled is recorded. 1. What is the sample space for this experiment? 2. Give the event E that an odd number is recorded. 3. Give the event F that a number greater than 5 is recorded. 4. Are E and F mutually exclusive? 4 Example: A company is working on a new lightbulb. Each experiment consists of installing the lightbulb and recording the number of hours until it burns out. 1. What is the sample space? 2. What is the event E, that the lightbulb lasts between 80 and 100 hours? 5