Question

Suppose you play a coin toss game in which you win​ $1 if a head appears and lose​ $1 if a tail appears. In the first 100 coin​ tosses, heads comes up 34 times and tails comes up 66 times. Answer parts​ (a) through​ (d) below. a. What percentage of times has heads come up in the first 100​ tosses? 34​% What is your net gain or loss at this​ point? Select the correct choice and fill in the answer box to complete your choice. A. You have lost ​$ 32. ​(Type an​ integer.) B. You have gained ​$ nothing. ​(Type an​ integer.) b. Suppose you toss the coin 200 more times​ (a total of 300​ tosses), and at that point heads has come up 37​% of the time. Is this change in the percentage of heads consistent with the law of large​ numbers? Explain. A. The change is consistent with the law of large numbers​ because, as the number of trials​ increases, the proportion should grow closer to​ 50%. B. The change is consistent with the law of large numbers. Because the percentage is low the first 100​ trials, it has to be higher the next 200 trials to even out. C. The change is not consistent with the law of large numbers​ because, as the number of trials​ increases, the proportion should grow closer to​ 50%. D. The change is not consistent with the law of large​ numbers, because the trials are not independent. What is your net gain or loss at this​ point? Select the correct choice and fill in the answer box to complete your choice. A. You have gained ​$ nothing. ​(Type an​ integer.) B. You have lost ​$ nothing. ​(Type an​ integer.) c. How many heads would you need in the next 100 tosses in order to break even after 400​ tosses? Is this likely to​ occur? Select the correct choice and fill in the answer box to complete your choice. A. You would need to toss nothing heads. This is likely because so few heads have been tossed so far. B. You would need to toss nothing heads. This is unlikely as it is far from the expected number of heads. C. You would need to toss nothing heads. This is likely because it is close to the expected number of heads. d. Suppose​ that, still behind after 400​ tosses, you decide to keep playing because you are due for a winning streak. Explain how this belief would illustrate the​ gambler's fallacy. A. This illustrates the​ gambler's fallacy because eventually there will be a winning streak. B. This illustrates the​ gambler's fallacy​ because, due to the law of large​ numbers, the probability of getting heads must now be more than 0.5. C. This illustrates the​ gambler's fallacy because the number of heads cannot be under​ 50% all the time. D. This illustrates the​ gambler's fallacy because the probability of getting heads is always 0.5.

Answer 1