MAT 221 Lecture_Unit-1 - Chapter 1 Introduction to...

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Chapter 1 Introduction to Probability Definition 1.0.1. An experiment is a procedure we perform (quite often hypothetical) that produces some result. Definition 1.0.2. An outcome is a possible result of an experiment. Definition 1.0.3. An event is a certain set of outcomes of an experiment. Definition 1.0.4. The sample space is the collection or set of all possible outcomes of an experiment. The letter S is used to designate the sample space, which is the universal set of outcomes of an experiment. A sample space is called discrete if it is a finite or a countably infinite set. It is called continuous or a continuum otherwise. Example 1.0.5. Experiment: Tossing two coins. Sample Space: S = { HH, HT, TH, TT } . Event=at least one Head; E = { HH, HT, TH } . Definition 1.0.6. Two events are mutually exclusive if they cannot occur at the same time. Example 1.0.7. S = { H, T } , E 1 = { H } , E 2 = { T } . Then events E 1 and E 2 are mutually exclusive. In this case, E 1 E 2 = . Definition 1.0.8 ( Classical definition of Probability ) . The probability of an event equals the ratio of its favorable outcomes to the total number of outcomes provided that all outcomes are equally likely. Definition 1.0.9. The probability of event E is given by P ( E ) = n ( E ) n ( S ) , where n ( S ) < and n ( · ) denotes the number of elements. Example 1.0.10. Find the probability of obtaining at least one head when two coins are tossed. 1
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CHAPTER 1. INTRODUCTION TO PROBABILITY Definition 1.0.11 ( Axioms of Probability ) . Let S be the sample space, E be an event. We assign to each event E a number P ( E ), which we call the probability of the event E . This number is so chosen as to satisfy the following three conditions: Axiom 1: 0 P ( E ) 1. Axiom 2: P ( S ) = 1. Axiom 3: P ( E F ) = P ( E ) + P ( F ), if events E and F are mutually exclusive. Example 1.0.12. Show that P ( n i =1 E i ) = n summationdisplay i =1 P ( E i ), where E i E j = when i negationslash = j . Example 1.0.13. Find the probability of rolling an even number when a dice is rolled. Proposition 1.0.14. Let E and F be events and E c be the compliment of E . Then 1. P ( E c ) = 1 P ( E ) . 2. If E F , then P ( E ) P ( F ) . 3. P ( E F ) = P ( E ) + P ( F ) P ( E F ) . 4. P ( i =1 E i ) = summationdisplay i =1 P ( E i ) , where E 1 , E 2 , . . . are any sequence of mutually exclusive events (i.e., events for which E i E j = when i negationslash = j ).
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