ACTIVITY 6.2 CHOICES 719 11. a. Determine the complement of event A in the roll-a-die experiment. b. Describe what portion of the Venn diagram above represents the complement of A. SUMMARY Activity 6.2 1. The multiplication principle of counting says that if some choice can be made in M ways and a subsequent choice can be made in N ways, then there are M times N ways these choices can be made in succession. 2. A tree diagram displays all possible outcomes for a sequence of choices, one outcome for each branch of the tree. 3. The sample space of a probability distribution is the collection of all possible outcomes. 4. The sum of the probability that an event A will occur and the probability that the event will not occur is 1. The event "not A," denoted by A, is called the complement of A. Stated symbolically, P(A) + P(A~) = 1 or P(A) = 1 - P(A). EXERCISES Activity 6.2 \\\v?i 1. Phone numbers consist of a three-digit area code followed by seven digits. If the area code must have a 0 or I for the second digit, and neither the area code nor the sevendigit number can start with 0 or 1, how many different phone numbers are possible? 2. You have four sweaters, five pairs of pants, and three pairs of shoes. How many different combinations can you make, wearing one of each? 3. If you flip a coin ten time.s, how many different sequences of heads and tails are possible? 4. If you roll a die three times, how many different sequences are possible?
720 CHAPTER 6 PROBABILITY MODELS 5. You want to order a triple-scoop ice cream cone, with a different flavor for each scoop. If there are 23 flavors available, how many different cones are possible? 6. In a single experiment, a die is tossed and a spinner with the letters A, B, and C is spun. Each letter is equally likely. a. Determine the number of possible outcomes. b. Construct a tree diagram to help determine the resulting sample space. c. Determine the probability of getting a 2 on the die and a B on the spinner. d. Determine the probability of getting a B. e. Determine the probability of getting a 2 or a B. 7. A computer is programmed to generate a sequence of three digits, where each digit is either 0 or 1, and each of these is equally likely to occur. a. Construct a tree diagram that shows all possible three-digit sequences of 0s and Is.
ACTIVITY 6.2 CHOICES 721 b. What is the probability that a sequence will contain exactly one 0? c. What is the probability that a sequence will contain at least one 0? 8. Two teams are playing a best-of-seven series. For each game there are only two possible outcomes: One team wins or the other team wins. Construct a tree diagram showing all the possible outcomes for such a series. As soon as one team wins a fourth game, the series ends (and that branch of the tree ends also). 9. You roll two fair dice. Use the sample space you recorded in Problem 5 of the activity to determine the probabilities of the following events. a. P(a sum of 7) b. P(both dice show an even number) c. P(one die is even, the other odd) d. P(one die is 1 more than the other) e. P(the sum is less than 6) I'. P(thc product of the two numbers on the dice is even) 10. You have decided to purchase a new computer system. After researching the cost and features of different models of computers, printers, and monitors, you have narrowed your choices to the following: Computer: Printer: Monitor: Dell, Compaq, and IBM Epson, Hewlett-Packard Omega, Toshiba a. Determine the number of possible computer system combinations that you can purchase.
722 CHAPTER 6 PROBABILITY MODELS b. Construct a tree diagram and list the possible computer systems that can be purchased. c. Determine the probability that a Dell computer is part of the system purchased. Assume that all outcomes are equally likely. d. What is the probability that a Dell computer is not part of the computer system you select? e. Determine the probability that an Epson printer is part of the computer package purchased. f. Determine the probability that taahe system purchased contained an IBM computer and Omega monitor. 11. Two thumbtacks are dropped, one after another, onto a hard surface. Assume that the thumbtack can only land point up or point down. Let U represent the event that the thumbtack will land point up. D represent the event that the thumbtack will land point down.
ACTIVITY 6.2 CHOICES 723 a. Are the events U and D equally likely? Explain. b. Design and conduct an experiment involving dropping thumbtacks. Use the results to determine the experimental probability that a thumbtack will land point up when dropped and the experimental probability that the thumbtack will land point down.