Chapter 6: Probability and Simulation: The Study of Randomness Review Sheet

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1 Review Sheet 1. I toss a penny and observe whether it lands heads up or tails up. Suppose the penny is fair; that is, the probability of heads is ½ and the probability of tails is ½. This means A) that every occurrence of a head must be balanced by a tail in one of the next two or three tosses. B) that if I flip the coin many, many times, the proportion of heads will be approximately ½, and this proportion will tend to get closer and closer to ½ as the number of tosses increases. C) that regardless of the number of flips, half will be heads and half tails. D) generally, the flips will alternate between heads and tails. E) all of the above. 2. A phenomenon is observed many, many times under identical conditions. The proportion of times a particular event A occurs is recorded. This proportion represents A) the probability of the event A. D) the proportionality of the event A. B) the distribution of the event A. E) the variance of the event A. C) the correlation of the event A. 3. If the individual outcomes of a phenomenon are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions, we say the phenomenon is A) random. B) predictable. C) deterministic. D) probabilistic. E) none of the above. 4. Suppose we have a loaded die that gives the outcomes 1 6 according to the following probability distribution: X P(X) Note that for this die all outcomes are not equally likely, as they would be if the die were fair. If this die is rolled 6000 times, the number of times we get a 2 or a 3 should be about A) B) C) D) E) The answer cannot be determined since the probabilities are only approximate. 5. Suppose we roll a red die and a green die. Let A be the event that the number of spots showing on the red die is 3 or less and B be the event that the number of spots showing on the green die is more than 3. The events A and B are A) disjoint. B) complements. C) independent. D) reciprocals. E) dependent.

2 6. In a particular game, a fair die is tossed. If the number of spots showing is either 4 or 5, you win $1; if the number of spots showing is 6, you win $4; and if the number of spots showing is 1, 2, or 3, you win nothing. If it costs you $1 to play the game, the probability that you win more than the cost of playing is A) 0. B) 1/6. C) 1/3. D) 2/3. E) 5/6. 7. I select two cards from a deck of 52 cards and observe the color of each (26 cards in the deck are red and 26 are black). Which of the following is an appropriate sample space S for the possible outcomes? A) S = {red, black}. B) S = {(red, red), (red, black), (black, red), (black, black)}, where, for example, (red, red) stands for the event the first card is red and the second card is red. C) S = {0, 1, 2}. D) All of the above. E) The results will vary since the cards are drawn without replacement. 8. A game consists of drawing three cards at random from a deck of playing cards. You win $3 for each red card that is drawn. It costs $2 to play. For one play of this game, the sample space S for the net amount you win (after deducting the cost of play) is A) S = {$0, $1, $2, $3} D) S = { $2, $3, $6, $9} B) S = { $6, $3, $0, $6} E) S = {$0, $3, $6, $9} C) S = { $2, $1, $4, $7} 9. Event A occurs with probability 0.3. If event A and B are disjoint, then A) P(B) 0.3. B) P(B) 0.3. C) P(B) 0.7. D) P(B) 0.7. E) P(B) = Event A occurs with probability 0.2. Event B occurs with probability 0.8. If A and B are disjoint (mutually exclusive), then A) P(A and B) = D) P(A or B) = B) P(A and B) = E) P(A or B) = 1. C) P(A and B) = 1. 83

3 11. Suppose there are three cards in a deck, one marked with a 1, one marked with a 2, and one marked with a 5. You draw two cards at random and without replacement from the deck of three cards. The sample space S = {(1, 2), (1, 5), (2, 5)} consists of these three equally likely outcomes. Let X be the sum of the numbers on the two cards drawn. Which of the following is the correct set of probabilities for X? A) X P(X) 1/3 1/3 1/3 B) X P(X) 1/3 1/3 1/3 C) X P(X) 3/16 6/16 7/16 D) X P(X) 1/4 1/2 1/4 E) X P(X) 1/4 1/2 1/4 Use the following to answer questions 12: A stack of four cards contains two red cards and two black cards. I select two cards, one at a time, and do not replace the first card before selecting the second card. 12. Referring to the information above, consider the events A = the first card selected is red B = the second card selected is black The events A and B are A) independent. B) disjoint. C) complements. D) reciprocals. E) none of the above. Use the following to answer questions 13 through 14: Ignoring twins and other multiple births, assume that babies births at a hospital represent independent events, with the probability that a baby is a boy and the probability that a baby is a girl both being equal to Referring to the information above, the probability that the next three babies are all of the same sex is A) 1. B) C) D) E) Referring to the information above, the probability that at least one of the next three babies is a boy is A) B) C) D) E) Suppose that A and B are two independent events with P(A) = 0.3 and P(B) = 0.3. P(A B) is A) B) C) D) E)

4 16. Suppose that A and B are two independent events with P(A) = 0.2 and P(B) = 0.4. P(A B C ) is A) B) C) D) E) In a certain town 60% of the households own a cellular phone, 40% own a pager, and 20% own both a cellular phone and a pager. The proportion of households that own a cellular phone but not a pager is A) 20%. B) 30%. C) 40%. D) 50%. E) 80%. Use the following to answer questions 18 through 21: A system has two components that operate in parallel, as shown in the diagram below. Since the components operate in parallel, at least one of the components must function properly if the system is to function properly. The probabilities of failures for the components 1 and 2 during one period of operation are.20 and.03, respectively. Let F denote the event that component 1 fails during one period of operation and G denote the event that component 2 fails during one period of operation. Component failures are independent. 18. The event corresponding to the system failing during one period of operation is A) F G. B) F G. C) (F G) c. D) F c G c. E) F c G c. 19. The event corresponding to the system functioning properly during one period of operation is A) F G. B) (F G) c. C) F c G c. D) F c G c. E) F G. 20. The probability that the system functions properly during one period of operation is A) B) C) D) E) The probability that the system fails during one period of operation is A) B) C) D) E) Use the following to answer questions 22-23: An event A will occur with probability 0.5. An event B will occur with probability 0.6. The probability that both A and B will occur is

5 22. Referring to the information above, the conditional probability of A given B A) is 0.3. B) is 0.2. C) is 1/6. D) is 0.1. E) cannot be determined from the information given. 23. Referring to the information above, we may conclude A) that events A and B are independent. D) that events A and B are disjoint. B) that events A and B are complements. E) none of the above. C) that either A or B always occurs. 24. Event A occurs with probability 0.8. The conditional probability that event B occurs given that A occurs is 0.5. The probability that both A and B occur A) is 0.3. B) is 0.4. C) is 0.5. D) is 0.8. E) cannot be determined from the information given. 25. Event A occurs with probability 0.3 and event B occurs with probability 0.4. If A and B are independent, we may conclude that A) P(A and B) = D) P(A or B) = B) P(A B) = 0.3. E) all of the above. C) P(B A) = The probability of a randomly selected adult having a rare disease for which a diagnostic test has been developed is The diagnostic test is not perfect. The probability the test will be positive (indicating that the person has the disease) is 0.99 for a person with the disease and 0.02 for a person without the disease. The proportion of adults for which the test would be positive is A) B) C) D) E) In question 38, if a randomly selected person is tested and the result is positive, the probability the individual has the disease is A) B) C) D) E)

6 Answer Key 1. 1 B 2. 2 A 4. 3 A 6. 4 C 7. 5 C 8. 6 B 9. 7 B C C E B E C E B B C A C A E C C B E D E 87

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