Chapter 7 Probability

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1 Chapter 7 Probability Section 7.1 Experiments, Sample Spaces, and Events Terminology Experiment An experiment is an activity with observable results. The results of an experiment are called outcomes of the experiment. Examples Tossing a coin and observing whether it falls heads or tails Rolling a die and observing which of the numbers 1, 2, 3, 4, 5, or 6 shows up Testing a spark plug from a batch of 100 spark plugs and observing whether or not it is defective Sample Point, Sample Space, and Event Sample point: An outcome of an experiment Sample space: The set consisting of all possible sample points of an experiment Event: A subset of a sample space of an experiment Example: Describe the sample space associated with the experiment of tossing a coin and observing whether it falls heads or tails. What are the events of this experiment? Union of Two Events The union of two events E and F is the event E F. Thus, the event E F contains the set of outcomes of E and/or F. Intersection of Two Events The intersection of two events E and F is the event E F. Thus, the event E F contains the set of outcomes common to E and F. Complement of an Event The complement of an event E is the event E c. Thus, the event E c is the set containing all the outcomes in the sample space S that are not in E. 1 P a g e

2 Example: Consider the experiment of rolling 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 E F. Interpret your results. Compute E F. Interpret your results. Compute F c. Interpret your results Mutually Exclusive Events E and F are mutually exclusive if E F = Ø Example: An experiment consists of tossing a coin three times and observing the resulting sequence of heads and tails. Describe the sample space S of the experiment. Determine the event E that exactly two heads appear. Determine the event F that at least one head appears. 2 P a g e

3 Example: Applied Example: Movie Attendance The manager of a local cinema records the number of patrons attending a first-run movie screening. The theatre has a seating capacity of 500. Determine an appropriate sample space for this experiment. Describe the event E that fewer than 50 people attend the screening. Describe the event F that the theatre is more than half full at the screening. 3 P a g e

4 Section 7.2 Definition of Probability Probability of an Event in a Uniform Sample Space If S = {s 1, s 2,, s n } is the sample space for an experiment in which the outcomes are equally likely, then we assign the probabilities to each of the outcomes s 1, s 2,, s n. Example: Tossing a Coin If a single fair coin is tossed, find the probability that it will land heads up. The sample space S = {h, t}, and the event whose probability we seek is E = {h}. P(heads) = P(E) = 1/2. Since no coin flipping was actually involved, the desired probability was obtained theoretically. Theoretical Probability Formula If all outcomes in a sample space S are equally likely, and E is an event within that sample space, then the theoretical probability of the event E is given by number of favorable outcomes ne ( ) PE ( ). total number of outcomes ns ( ) Example: A fair die is rolled, and the number that falls uppermost is observed. Determine the probability distribution for the experiment. Finding the Probability of Event E 1. Determine a sample space S associated with the experiment. 2. Assign probabilities to the simple events of S. 3. If E = {s 1, s 2,, s n } where {s 1 }, {s 2 }, {s 3 },, {s n } are simple events, then P(E) = P(s 1 ) + P(s 2 ) + P(s 3 ) + + P(s n ) If E is the empty set, Ø, then P(E) = 0. 4 P a g e

5 Example: Flipping a Cup A cup is flipped 100 times. It lands on its side 84 times, on its bottom 6 times, and on its top 10 times. Find the probability that it will land on its top. From the experiment it appears that P(top) = 10/100 = 1/10. This is an example of experimental, or empirical probability. Empirical Probability Formula If E is an event that may happen when an experiment is performed, then the empirical probability of event E is given by number of times event E occurred PE ( ). number of times the experiment was performed Example: Card Hands There are 2,598,960 possible hands in poker. If there are 36 possible ways to have a straight flush, find the probability of being dealt a straight flush. 36 P(straight flush) ,598,960 Example: Gender of a Student A school has 820 male students and 835 female students. If a student from the school is selected at random, what is the probability that the student would be a female? number of female students P(female) total number of students P a g e

6 Section 7.3 Rules of Probability Properties of the Probability Function Property 1. P(E) 0 for every E. Property 2. P(S) = 1. Property 3. If E and F are mutually exclusive (E F = Ø), then P(E F) = P(E) + P(F) Example: Applied Example: SAT Verbal Scores The superintendent of a metropolitan school district has estimated the probabilities associated with the SAT verbal scores of students from that district. The results are shown in the table below. If a student is selected at random, find the probability that his or her SAT verbal score will be More than 400. Less than or equal to 500. Greater than 400 but less than or equal to 600. Score, x Probability x > < x < x < x < x x P a g e

7 Addition Rule Property 4. If E and F are any two events of an experiment, then P(E F) = P(E) + P(F) P(E F) Example: A card is drawn from a shuffled deck of 52 playing cards. What is the probability that it is an ace or a spade? Example: Applied Example: Quality Control The quality-control department of Vista Vision, manufacturer of the Pulsar plasma TV, has determined from records obtained from the company s service centers that 3% of the sets sold experience video problems, 1% experience audio problems, and 0.1% experience both video and audio problems before the expiration of the warranty. Find the probability that a plasma TV purchased by a consumer will experience video or audio problems before the warranty expires. 7 P a g e

8 Rule of Complements Property 5. If E is an event of an experiment and E c denotes the complement of E, then P(E c ) = 1 P(E) Example: Applied Example: Warranties What is the probability that a Pulsar plasma TV (from the last example) bought by a consumer will not experience video or audio problems before the warranty expires? 8 P a g e

9 Section 7.4 Use of Counting Techniques in Probability 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, PE ( ) Number of outcomes in Number of outcomes in E n ( E ) S n ( S ) Events Involving Not and Or Properties of Probability Let E be an event from the sample space S. That is, E is a subset of S. Then the following properties hold PE ( ) 1 (The probability of an event is between 0 and 1, inclusive.) 2. P( ) 0 (The probability of an impossible event is 0.) 3. PS ( ) 1 (The probability of a certain event is 1.) Example: Rolling a Die When a single fair die is rolled, find the probability of each event. a) the number 3 is rolled b) a number other than 3 is rolled c) the number 7 is rolled d) a number less than 7 is rolled The outcome for the die has six possibilities: {1, 2, 3, 4, 5, 6}. 1 a) P(3) 6 5 b) P(not 3) 6 c) P(7) 0 d) P(less than 7) 1 9 P a g e

10 Events Involving Not The table on the next slide shows the correspondences that are the basis for the probability rules developed in this section. For example, the probability of an event not happening involves the complement and subtraction. Correspondences Set Theory Logic Arithmetic Operation or Connective (Symbol) Operation or Connective (Symbol) Operation or Connective (Symbol) Complement Not Subtraction Union Or Addition Intersection And Multiplication Probability of a Complement The probability that an event E will not occur is equal to one minus the probability that it will occur. P(not E) P( S) P( E) 1 PE ( ) So we have and P( E) P E 1 P( E) 1 P( E). Example: Complement When a single card is drawn from a standard 52-card deck, what is the probability that is will not be an ace? P(not an ace) 1 P(ace) P a g e

11 Events Involving Or Probability of one event or another should involve the union and addition. Mutually Exclusive Events Two events A and B are mutually exclusive events if they have no outcomes in common. (Mutually exclusive events cannot occur simultaneously.) Addition Rule of Probability (for A or B) If A and B are any two events, then P( A or B) P( A) P( B) P( A and B). If A and B are mutually exclusive, then P( A or B) P( A) P( B). Example: Probability Involving Or When a single card is drawn from a standard 52-card deck, what is the probability that it will be a king or a diamond? P(king or diamond) P(K) P(D) P(K and D) Example: Probability Involving Or If a single die is rolled, what is the probability of a 2 or odd? These are mutually exclusive events. P(2 or odd) P(2) P(odd) P a g e

12 Section 7.5 Conditional Probability and Independent Events Conditional Probability of an Event If A and B are events in an experiment and P(A) ¹ 0, then the conditional probability that the event B will occur given that the event A has already occurred is P ( A B P( B A) ) PA ( ) Conditional Probability Sometimes the probability of an event must be computed using the knowledge that some other event has happened (or is happening, or will happen the timing is not important). This type of probability is called conditional probability. The probability of event B, computed on the assumption that event A has happened, is called the conditional probability of B, given A, and is denoted P(B A). Example: Selecting From a Set of Numbers From the sample space S = {2, 3, 4, 5, 6, 7, 8, 9}, a single number is to be selected randomly. Given the events A: selected number is odd, and B selected number is a multiple of 3. find each probability. a) P(B) b) P(A and B) c) P(B A) a) B = {3, 6, 9}, so P(B) = 3/8 b) P(A and B) = {3, 5, 7, 9} {3, 6, 9} = {3, 9}, so P(A and B) = 2/8 = 1/4 c) The given condition A reduces the sample space to {3, 5, 7, 9}, so P(B A) = 2/4 = 1/2 Conditional Probability Formula The conditional probability of B, given A, and is given by P( A B) P( A and B) P( B A). P( A) P( A) 12 P a g e

13 Example: Probability in a Family Given a family with two children, find the probability that both are boys, given that at least one is a boy. Define S = {gg, gb, bg, bb}, A = {gb, bg, bb}, and B = {bb}. P( A B) 1/ 4 1 P( B A). PA ( ) 3/ 4 3 Independent Events Two events A and B are called independent events if knowledge about the occurrence of one of them has no effect on the probability of the other one, that is, if P(B A) = P(B), or equivalently P(A B) = P(A). Example: Checking for Independence A single card is to be drawn from a standard 52-card deck. Given the events A: the selected card is an ace B: the selected card is red a) Find P(B). b) Find P(B A). c) Determine whether events A and B are independent a. PB ( ) P( A B) 2/ b. P( B A) = = = =. PA ( ) 4/ c. Because P(B A) = P(B), events A and B are independent. 13 P a g e

14 Events Involving And If we multiply both sides of the conditional probability formula by P(A), we obtain an expression for P(A and B). The calculation of P(A and B) is simpler when A and B are independent. Multiplication Rule of Probability If A and B are any two events, then P( A and B) P( A) P( B A). If A and B are independent, then P( A and B) P( A) P( B). Example: Selecting From an Jar of Balls Jeff draws balls from the jar below. He draws two balls without replacement. Find the probability that he draws a red ball and then a blue ball, in that order. 4 red 3 blue 2 yellow P( R1 and B2 ) P( R1) P( B2 R1) P a g e

15 Example: Selecting From an Jar of Balls Jeff draws balls from the jar below. He draws two balls, this time with replacement. Find the probability that he gets a red and then a blue ball, in that order. 4 red 3 blue 2 yellow Because the ball is replaced, repetitions are allowed. In this case, event B 2 is independent of R 1. P( R1 and B2 ) P( R1) P( B2 ) P a g e

16 Example: Applied Example: Color Blindness In a test conducted by the U.S. Army, it was found that of 1000 new recruits (600 men and 400 women), 50 of the men and 4 of the women were red-green color-blind. Given that a recruit selected at random from this group is red-green color-blind, what is the probability that the recruit is a male? 16 P a g e

17 Product Rule P ( A B ) P ( A ) P ( B A ) Example: Two cards are drawn without replacement from a well-shuffled deck of 52 playing cards. What is the probability that the first card drawn is an ace and the second card drawn is a face card? 17 P a g e

18 Independent Events If A and B are independent events, then P ( A B ) P ( A ) and P ( B A ) P ( B ) Test for the Independence of Two Events Two events A and B are independent if and only if P ( A B ) P ( A ) P ( B ) Example: Consider the experiment consisting of tossing a fair coin twice and observing the outcomes. Show that obtaining heads on the first toss and tails on the second toss are independent events. 18 P a g e

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