Sts

1. I toss a penny and observe whether it lands heads up or tails up. Suppose the penny is fair, i.e., the probability of heads is 1/2 and the probability of tails is 1/2. This means that A. every occurrence of a head must be balanced by a tail in one of the next two or three tosses. B. if I flip the coin 10 times, it would be almost impossible to obtain 7 heads and 3 tails. C. if I flip the coin many, many times the proportion of heads will be approximately 1/2, and this proportion will tend to get closer and closer to 1/2 as the number of tosses increases. D. regardless of the number of flips, half will be heads and half tails. E. all of the above.

1. C

12. A basketball player makes 2/3 of his free throws. To simulate a single free throw, which of the following assignments of digits to making a free throw are appropriate? I. 0 and 1 correspond to making the free throw and 2 corresponds to missing the free throw. II. 01, 02, 03, 04, 05, 06, 07, and 08 correspond to making the free throw and 09, 10, 11, and 12 correspond to missing the free throw. III. Use a die and let 1, 2, 3, and 4 correspond to making a free throw while 5 and 6 correspond to missing a free throw. A. I only B. II only C. III only D. I and III E. I, II, and III

12. E

14. Use Scenario 5-1. Based on your simulation, the estimated probability of winning $2 in this game is A. 1/4. B. 5/15. C. 7/15. D. 9/15. E. 7/11.

14. E

2. 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. uniform. D. probable. E. normal.

2. A

20. 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) (B) X P(X) (C) X P(X) (D) X P(X) (E) X P(X) 1 1/3 3 1/3 3 3/16 3 1/4 1 1/4 2 1/3 6 1/3 6 6/16 6 1/2 2 1/2 5 1/3 7 1/3 7 7/16 7 1/2 5 1/2 A. A B. B C. C D. D E. E

20. B

6. A basketball player makes 160 out of 200 free throws. We would estimate the probability that the player makes his next free throw to be A. 0.16. B. 50-50; either he makes it or he doesn't. C. 0.80. D. 1.2. E. 80.

6. C

7. In probability and statistics, a random phenomenon is A. something that is completely unexpected or surprising B. something that has a limited set of outcomes, but when each outcome occurs is completely unpredictable. C. something that appears unpredictable, but each individual outcome can be accurately predicted with appropriate mathematical or computer modeling. D. something that is unpredictable from one occurrence to the next, but over the course of many occurrences follows a predictable pattern E. something whose outcome defies description.

7. D

10. You want to use simulation to estimate the probability of getting exactly one head and one tail in two tosses of a fair coin. You assign the digits 0, 1, 2, 3, 4 to heads and 5, 6, 7, 8, 9 to tails. Using the following random digits to execute as many simulations as possible, what is your estimate of the probability? 19226 95034 05756 07118 A. 1/20 B. 1/10 C. 5/10 D. 6/10 E. 2/3

10. D

11. A box has 10 tickets in it, two of which are winning tickets. You draw a ticket at random. If it's a winning ticket, you win. If not, you get another chance, as follows: your losing ticket is replaced in the box by a winning ticket (so now there are 10 tickets, as before, but 3 of them are winning tickets). You get to draw again, at random. Which of the following are legitimate methods for using simulation to estimate the probability of winning? I. Choose, at random, a two-digit number. If the first digit is 0 or 1, you win on the first draw; If the first digit is 2 through 9, but the second digit is 0, 1, or 2, you win on the second draw. Any other two-digit number means you lose. II. Choose, at random, a one-digit number. If it is 0 or 1, you win. If it is 2 through 9, pick a second number. If the second number is 8, 9, or 0, you win. Otherwise, you lose. III. Choose, at random, a one-digit number. If it is 0 or 1, you win on the first draw. If it is 2, 3, or 4, you win on the second draw; If it is 5 through 9, you lose. A. I only B. II only C. III only D. I and II E. I, II, and III

11. D

13. A basketball player makes 75% of his free throws. We want to estimate the probability that he makes 4 or more frees throws out of 5 attempts (we assume the shots are independent). To do this, we use the digits 1, 2, and 3 to correspond to making the free throw and the digit 4 to correspond to missing the free throw. If the table of random digits begins with the digits below, how many free throw does he hit in our first simulation of five shots? 19223 95034 58301 A. 1 B. 2 C. 3 D. 4 E. 5 Scenario 5-1 To simulate a toss of a coin we let the digits 0, 1, 2, 3, and 4 correspond to a head and the digits 5, 6, 7, 8, and 9 correspond to a tail. Consider the following game: We are going to toss the coin until we either get a head or we get two tails in a row, whichever comes first. If it takes us one toss to get the head we win $2, if it takes us two tosses we win $1, and if we get two tails in a row we win nothing. Use the following sequence of random digits to simulate this game as many times as possible: 12975 13258 45144

13. E

15. Use Scenario 5-1. Based on your simulation, the estimated probability of winning nothing is A. 1/2. B. 2/11. C. 2/15. D. 6/15. E. 7/11.

15. B

16. The collection of all possible outcomes of a random phenomenon is called A. a census. B. the probability. C. a chance experiment D. the sample space. E. the distribution.

16. D

17. 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 = {(red, red), (red, black), (black, black)}, where, for example, (red, red) stands for the event "the first card is red and the second card is red." D. S = {0, 1, 2}. E. All of the above.

17. B

18. A basketball player shoots 8 free throws during a game. The sample space for counting the number she makes is A. S = any number between 0 and 1. B. S = whole numbers 0 to 8. C. S = whole numbers 1 to 8. D. S = all sequences of 8 hits or misses, like HMMHHHMH. E. S = {HMMMMMMM, MHMMMMMM, MMHMMMMM, MMMHMMMM, MMMMHMMM, MMMMMHMM, MMMMMMHM, MMMMMMMH}

18. B

19. 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} B. S = {-$6, -$3, $0, $6} C. S = { -$2, $1, $4, $7} D. S = { -$2, $3, $6, $9} E. S = {$0, $3, $6, $9}

19. C

3. When two coins are tossed, the probability of getting two heads is 0.25. This means that A. of every 100 tosses, exactly 25 will have two heads. B. the odds against two heads are 4 to 1. C. in the long run, the average number of heads is 0.25. D. in the long run two heads will occur on 25% of all tosses. E. if you get two heads on each of the first five tosses of the coins, you are unlikely to get heads the fourth time.

3. D

4. If I toss a fair coin 5000 times A. and I get anything other than 2500 heads, then something is wrong with the way I flip coins. B. the proportion of heads will be close to 0.5 C. a run of 10 heads in a row will increase the probability of getting a run of 10 tails in a row. D. the proportion of heads in these tosses is a parameter E. the proportion of heads will be close to 50.

4. B

5. You read in a book on poker that the probability of being dealt three of a kind in a five-card poker hand is 1/50. What does this mean? A. If you deal thousands of poker hands, the fraction of them that contain three of a kind will be very close to 1/50. B. If you deal 50 poker hands, then one of them will contain three of a kind. C. If you deal 10,000 poker hands, then 200 of them will contain three of a kind. D. A probability of 0.02 is somebody's best guess for a probability of being dealt three of a kind. E. It doesn't mean anything, because 1/50 is just a number.

5. A

8. You are playing a board game with some friends that involves rolling two six-sided dice. For eight consecutive rolls, the sum on the dice is 6. Which of the following statements is true? A. Each time you roll another 6, the probability of getting yet another 6 on the next roll goes down. B. Each time you roll another 6, the probability of getting yet another 6 on the next roll goes up. C. You should find another set of dice: eight consecutive 6's is impossible with fair dice. D. The probability of rolling a 6 on the ninth roll is the same as it was on the first roll. E. None of these statements is true.

8. D

9. A poker player is dealt poor hands for several hours. He decides to bet heavily on the last hand of the evening on the grounds that after many bad hands he is due for a winner. A. He's right, because the winnings have to average out. B. He's wrong, because successive deals are independent of each other. C. He's right, because successive deals are independent of each other. D. He's wrong, because he's clearly on a "cold streak." E. Whether he's right or wrong depends on how many bad hands he's been dealt so far.

9. B