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Section 7.1 Experiments, Sample Space, and Events Terminology Experiment is an activity with observable results. Outcomes of the experiment are the results of the experiment. Sample Point is an outcome of an experiment. Sample Space is the set of consisting of all possible sample points of an experiment. Event is a subset of a sample space of an experiment. If E and F are two events, then, the union of two events, the event E F, the intersection of two events, the event E F, the complement of an event E, the event E C , and the mutually exclusive events E and F if E F = . Example 1: Consider the experiment of casting a die and observing the number that falls uppermost. Let S {1, 2, 3, 4, 5, 6} denote the sample space of the experiment and E = {2, 4, 6} and F = {1, 3} be events of this experiment. Compute a) E F, b) E F, and E C . Example 2: An experiment consists of tossing a coin three times and observing the resulting sequence of “heads” and “tails”. a) Describe the sample space S of the experiment. b) Determine the event E that exactly two heads appear c) Determine the event F that at least one head appears. Example 3: An experiment consists of casting a pair of dice and observing the number that fall s uppermost each die. a) Describe an appropriate sample space of S for this Experiment. b) Determine the events E1, E2, ......., E12 that the sum of the numbers falling uppermost is 1, 2, 3, 4... 12, respectively. Example 4 An experiment consists of studying the composition of a three-child family in which children were born at different times. a) Describe an appropriate sample space S for this experiment. b) Describe the event E that there are two girls and a boy in the family c) Describe the event F that oldest child is a girl. d) Describe the event G that oldest child is a girl and the youngest child is a boy. Example 5 The Ever-Brite Battery Company is developing high amperage, high –capacity battery as a source for powering electric cars. The battery is tested by installing it in a prototype electric car and running the car with a fully charged battery on a test track at a constant speed of 55 mph until the car runs out of power. The distance covered by the car is then observed. a) What is the sample space for this experiment? b) Describe the event E that the driving rang under test conditions is less than 150 miles c) Describe the event F that the driving range is between 200 and 250 miles, inclusive. Section 7.2 Definition of Probability If S is a finite sample space with n outcomes, which is S= { s1, s2 ... sn }, then, the events { s1 }, { s2 } ... { sn } which consists of exactly one point, are called simple or elementary, events of the experiment. Simple events are mutually exclusive. The probability Distribution: Simple Event { s1 } { s2 } _ _ _ { sn } Probability P ( s1 ) P ( s2 ) _ _ _ P ( sn ) The Probability Function: 1. 0<= P ( si ) <=1 (i=1, 2,...n) 2. P( s1 )+P( s2 )+...+P( sn )=1 3. P( { si } { sj } )= P( si ) +P( sj ) i j (i=1,2..n;j=1,2..n) Uniform Sample Spaces: If S = { s1, s2 ... sn } is the sample space for an experiment which the outcomes are equally likely, then assign the probabilities P( s1 ) =P( s2 ) =...= P( sn )= 1n to each of the simple events s1, s2 ... sn Computing the probability of an event in a uniform sample space: Let S be a uniform sample space and let E be any event. Then, of favorable outcomes in E n( E ) P (E) = Number Number of favorable outcomes in S n( S ) Example 1 A fair die is cast and the number that falls uppermost is observed. Determine the probability distribution for the experiment. Example 2 Refer to the table below which is tests involving 200 car test runs. Each run was made with fully charged battery. Distance Covered in Miles(x) 0<=x<=50 50<x<=100 100<x<=150 150<x<=200 200<x<=250 250>x Frequency of Occurrence 4 10 30 100 40 16 a) Describe an appropriate sample space for this experiment b) Find the probability distribution for this experiment. Find a Probability of an Event E 1. Determine a sample space S associated with the experiment. 2. Assign probabilities to the sample events of S. 3. If E = { s1, s2 ... sn } where { s1 }, { s2 } ... { sn } are simple events, then, P (E) = P ( s1 ) +P ( s2 ) +...+P ( sn ) If E is empty set, P (E) =0 Example 3 If a ball is selected at random from an urn containing three red balls, two white balls and five blue balls what is the probability that it will be white ball? Example 4 A pair of fair dice is cast. What is the probability that a) The sum of the numbers shown uppermost is less than 5? b) At least one 6 is cast? Example 5 Let S = { s1, s2 , s3, s4, s5, s6 } be the sample space associated with an experiment having the following probability distribution Outcome Probability { s1 } { s2 } { s3 } { s4 } { s5 } 1 12 1 4 1 12 1 6 1 3 Find the probability of the event a) A = { s1, s2 } b) B = { s3, s2 , s5, s6 } c) C = S { s6 } 1 12 Section 7.3 Rules of Probability Rules: Let S be a sample space, and E and F be events 1. P(E)>=0, P(S) =1 2. If and F are mutually exclusive ( E F = ), then, P ( E F ) = P (E) + P (F) 3. Addition rule. If E and F are any two events of a experiment, then P ( E F ) = P (E) + P (F) – P ( E F ) 4. Rules of Complements P( E c ) = 1-P(E) Computing the probability of an event in a uniform sample space: Let S be a uniform sample space and let E be any event. Then, of favorable outcomes in E n( E ) P (E) = Number Number of favorable outcomes in S n( S ) Example 1: A card is drawn from a well-shuffled deck of 52 playing cards. What is the probability that it is an ace or a spade? Example 2: Let E and F be two events of an experiment with sample space S. Suppose P(E) = .2 and P(F) = .1 1. Assume that P ( E F ) = .05. Compute i. P ( E F ) ii. P( E c F c ) iii. P( E c F c ) 2. Assume that P ( E F ) = i. P ( E F ) ii. P ( E F ) iii. P( E c ) iv. P( E c F c ) Example 3: A={x | x is a graduate students who had learned Spanish at least one year in the Department of Foreign Languages}, B={x | x is a graduate students who had learned French at least one year in the Department of Foreign Languages}, C={x | x is x is a graduate students who had learned German at least one year in the Department of Foreign Languages}. n (A) =200, n (B) = 178, n(C) = 140, n ( A B) = 33, n( A C ) = 24, n(C B) = 18, and n( A B C) = 3 1. at least one year one of the three language: n ( A B C) =n(A)+n(B)+n(C) – n( ( A B) n( A C ) n(C B) n( A B C ) =200+178+140-33-24-18+3=446 P ( A B C ) =446/480=0.9291 2. at least one year exactly one of the three language : n( ( ( A B c C c ) ( A c C B c ) ( B C c A c )) = 146+130+101=377 P( ( ( A B C ) ( A 377/480=0.7854 c c c C B c ) ( B C c A c )) = U 480 A 3. None of these 30 brands: n(B ( A B C) ) =480-446=34 146 130 P( ( A B C) ) =34/480=0.0708 c c 21 3 15 101 C Example 4: Refer to the table below which is tests involving 200 car test runs. Each run was made with fully charged battery. Distance Covered in Frequency of Miles(x) 0<=x<=50 50<x<=100 100<x<=150 150<x<=200 200<x<=250 250>x Occurrence 4 10 30 100 40 16 c) Describe an appropriate sample space for this experiment d) Find the probability distribution for this experiment. e) What is the probability if the car is selected at random and the car’s running distance is less than 200 and greater than 100?