Math 1320 Chapter Seven Pack. Section 7.1 Sample Spaces and Events. Experiments, Outcomes, and Sample Spaces. Events. Complement of an Event

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1 Math 1320 Chapter Seven Pack Section 7.1 Sample Spaces and Events Experiments, Outcomes, and Sample Spaces An experiment is an occurrence with a result, or outcome, that is uncertain before the experiment takes place. The set of all possible outcomes is called the sample space for the experiment. Events Given a sample space S, an event E is a subset of S. The outcomes in E are called the favorable outcomes. We say that E occurs in a particular experiment if the outcome of that experiment is one of the elements of E - that is, if the outcome of the experiment is favorable. Complement of an Event The complement of an event E is the set of all outcomes not in E. Thus, the complement of E represents the event that E does not occur. Union of Events The union of the events E and F is the set of all outcomes in E or F (or both). Thus E F represents the event that E occurs or F occurs (or both). Intersection of Events The intersection of the events E and F is the set of all outcomes common to E and F. Thus, E F represents the event that both E and F occur. Mutually Exclusive Events If E and F are events, then E and F are said to be disjoint or mutually exclusive if E F is empty. (Hence, they have no outcomes in common.) Problem 1. Describe the sample space S of the experiment and list the elements of the given event. a) Three coins are tossed; the result is at most one head. b) Two distinguishable dice are rolled; the numbers add to 7. c) Two indistinguishable dice are rolled; both numbers are prime. d) A letter is chosen at random from the word Mozart; the letter is a vowel. e) A sequence of two different letters is randomly chosen from the digits 0 4; the first digit is larger than the second.

2 Problem 2. A packet of gummy candy contains four strawberry gums, four lime gums, two black current gums, and two orange gums. April May sticks her hand in and selects four at random. Complete the following sentences: a) The sample space is the set of b) April is particularly fond of combinations of two strawberry and two black currant gums. The event that April will get the combination she desires is the set of Problem 3. Let S be the sample space of the set of outcomes that result from tossing three coins. Suppose the three coins tossed are a dime, a nickel, and a quarter, in that order. So, for example, if the dime comes up heads, the nickel tails, and the quarter heads, the outcome would be (H, T, H). Let E be the set of outcomes where there the quarter comes up heads, and let F be the set of outcomes where exactly two coins come up tails. a) Express in words, and then list the elements of the set: E F. b) Express in words: E F c) Express in symbols: Either the quarter comes up heads, or two coins don t come up tails.

3 Section 7.2 Relative Frequency Estimated Probability When an experiment is performed a number of times, the estimated probability or relative frequency of an event E is the fraction of times that the event E occurs. If the experiment is performed N times and the event E occurs fr(e) times, then the estimated probability is given by P(E) = fr(e) N The number fr(e) is called the frequency of E. N, the number of times that the experiment Is performed, is called the number of trials or the sample size. Problem 1. The following table shows the frequency of outcomes when two indistinguishable coins were tossed 4000 times and the uppermost faces were observed. a) Determine the relative frequency distribution. b) What is the relative frequency that the second coin lands with heads up? c) What is the relative frequency that tails comes up at least once? Problem 2. The following table shows the crashworthiness ratings for 10 small SUVs. (3 = Good, 2 = Acceptable, 1 = Marginal, 0 = Poor) a) Find the relative frequency distribution for the experiment of choosing a small SUV at random and determining its frontal crash rating. b) What is the relative frequency that a randomly selected small SUV will have a crash test rating of Acceptable or better?

4 Problem 3. The following table shows the result of a survey of 100 authors by a publishing company. Compute the relative frequencies of the given events if an author as specified is chosen at random. a) An author is established and successful. b) An author is a new author. c) An author is unsuccessful. d) A successful author is established. e) An established author is successful.

5 Section 7.3 Probability and Probability Models Definition: A probability distribution is an assignment of a number P(s i ), the probability of s i, to each outcome of a finite sample space S = {s 1, s 2,, s n }. The probabilities must satisfy 1. 0 P(s i ) 1 2. P(s 1 ) + P(s 2 ) + + P(s n ) = 1. We find the probability of an event E, written P(E) by adding up the probabilities of the outcomes in E. If P(E) = 0, we call E and impossible event. The empty set event is always impossible, since something must happen. Definition: A probability model for a particular experiment is a probability distribution that predicts the relative frequency of each outcome if the experiment is performed a large number of times. Just as we think of relative frequency as estimated probability, we can think of modeled probability as theoretical probability. Probability Model for Equally Likely Outcomes: In an experiment in which all outcomes are equally likely, we model the experiment by taking the probability of an experiment to be Number of favorable outcomes P(E) = = n(e) Total number of outcomes n(s). Addition Principle: If A and B are any two events, then P(A B) = P(A) + P(B) P(A B). If A B =, we say that A and B are mutually exclusive, and we have P(A B) = P(A) + P(B). More Principles of Probability Distributions: The following rules hold for any sample space S and any event A: P(S) = 1 (The probability of something happening is 1) P( ) = 0 (The probability of nothing happening is 0) P(A ) = 1 P(A) (The probability of A not happening is 1 minus the probability of A) Problem 1. Complete the following probability distribution table and then calculate the stated probabilities. Outcome a b c d e Probability a. P({a, c, e}) b. P(E F), where E = {a, c, e} and F = {b, c, e} c. P(E ), where E is as in part (b) d. P(E F), where E and F are as in part (b).

6 Problem 2. Calculate the (modeled) probability P(E) using the given information, assuming that all outcomes are equally likely. a. n(s) = 8, n(e) = 4 b. S = {1, 3, 5, 7, 9}, E = {3, 7} Problem 3. An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the coins and dice are distinguishable and fair. a. Two coins are tossed; the result is one or more heads. b. Three coins are tossed; the result is more tails than heads. c. Two dice are rolled; the numbers add to 9. d. Two dice are rolled; one of the numbers is even, the other is odd. Problem 4. If two indistinguishable dice are rolled, what is the probability of the event {(4, 4), (2, 3)}? What is the corresponding event for a pair of distinguishable dice? Problem 5. A die is weighted in such a way that each of 1 and 2 is three times as likely to come up as each of the other numbers. Find the probability distribution. What is the probability of rolling an even number? Problem 6. Use the given information to find the indicated probability. a. P(A) =.3, P(B) =.4, P(A B) =.02. Find P(A B). b. A B =, P(B) =.8, P(A B) =.8. Find P(A). c. P(A B) = 1.0, P(A) =.6, P(A B) =.1. Find P(B). d. P(A) =.22. Find P(A ). Problem 7. Determine whether the information shown is consistent with a probability distribution. If not, say why. a. P(A) =.2; P(B) =.4; P(A B) =.2 b. P(A) =.2; P(B) =.4; P(A B) =.3

7 Problem 8. The following table shows the profile, by the math section of the SAT Reasoning Test, of admitted students at UCLA for the fall 2011 semester. Determine the probabilities of the following events (round answers to the nearest.01). a. An applicant had a Math SAT below 400. b. An applicant had a Math SAT of 700 or above and was admitted. c. An applicant did not have a Math SAT below 400. d. An applicant had a Math SAT of 700 or above or was admitted. Problem 9. According to a New York Times/CBS poll of March 2005, 49% agreed that Social Security taxes should be raised if necessary to keep the system afloat, and 43% agreed that it would be a good idea to invest part of their Social Security taxes on their own. What is the largest percentage of people who could have agreed with at least one of these statements? What is the smallest percentage of people who could have agreed with at least one of these statements? Problem 10. Lance the Wizard has been informed that tomorrow there will be a 50% chance of encountering the evil Myrmidons and a 20% chance of meeting up with the dreadful Balrog. Moreover, Hugo the Elf has predicted that there is a 10% chance of encountering both tomorrow. What is the probability that Lance will be lucky tomorrow and encounter neither the Myrmidons nor the Balrog? Homework: Read section 7.3, do #1, 3, 7, 13, 19, 21, 31, 71, 75

8 Section 7.4 Probability and Counting Techniques Modeling Probability: Equally Likely Outcomes In an experiment in which all outcomes are equally likely, the probability of an event E is given by P(E) = Number of favorable outcomes Total number of outcomes = n(e) n(s). Problem 1. Suzy is given a bag containing 4 red marbles, 3 green ones, 2 white ones, and 1 purple one. She grabs five of them. Find the probabilities of the following events, expressing each as a fraction in lowest terms. a) She has all the red ones. b) She has at least one white one. c) She has two red ones and one of each of the other colors. d) She has at most one green one. Problem 2. A test has three parts. Part A consists of eight true false questions, Part B consists of five multiple choice questions with five choices each, and Part C requires you to match five questions with five different answers one-to-one. Assuming that you make random guesses in filling out your answer sheet, what is the probability that you will earn 100% on the test? (Leave your answer as a formula.) Problem 3. Tyler and Gebriella are among seven contestants from which four semifinalists are to be selected at random. Find the probability that neither Tyler nor Gebriella is selected. Problem 4. You are asked to calculate the probability of being dealt various poker hands. (Recall that a poker player is dealt 5 cards at random from a standard deck of 52.) a) One pair: 2 cards with the same denomination and 3 cards with other denominations. b) Two pair: 2 cards with one denomination, 2 with another, and 1 with a third. c) Flush: Five cards of the same suit, but not a straight flush or a royal flush. Problem 5. In order to play the Mega Millions Lottery, we need to choose a ticket with five numbers from the set {1, 2,, 56}, and one number from the set {1, 2,, 46}. The order of the first five numbers does not matter. a) How many different tickets can we buy? b) How many tickets match all six winning numbers? c) We will win the Jackpot if we match all six winning numbers. Suppose we buy one ticket. What is the probability that we will win the Jackpot? d) We will win $10,000 if we match four of the five winning numbers from {1, 2,, 56}, and the one winning number from {1, 2,, 46}. How many different tickets will win $10,000? e) What is the probability that we will win $10,000 with one ticket?