20. In question 13 you should have found that roughly 90% of the probability was concentrated into 3 of the 5 macrostates. In question 18 you should have found that roughly 90% of the probability is concentrated into 5 of the 11 macrostates. Based on these results you can conclude that as you increase the number of coins in the system the probability. (a) becomes more spread out among a larger fraction of the macrostates (b) becomes more concentrated into a smaller fraction of the macrostates (c) remains about the same for a given fraction of the macrostates 21. Look back over your results for 3, 4, and 10 coins. Now suppose we did the same calculations for a sequence of 2000 coins (don't worry, we won't actually do this). Which of the following macrostates would have the highest probability? (a) 2000 H, 0 T (b) 1100 H, 900 T (c) 1001 H, 999 T (d) 1000 H, 1000 T 22. If we examined the probabilities for all of the macrostates in the system of 1000 coins we would find that the probability is (a) spread evenly among all of the macrostates (b) spread out among a large fraction of the macrostates, but with some macrostates having slightly higher probabilities than others (c) tightly concentrated within a small fraction of the macrostates centered on the most probable macrostate (d) entirely concentrated into the single most probable macrostate Experimenting with Our Model Now that you have a handle on multiplicities and entropy, let's perform a little "experiment" with our model. What would happen if we start with a row of 20 coins that are all showing heads and then we began flipping coins over at random? This may seem like a strange thing to investigate, but we will see later on that the behavior of this system provides us with a qualitative understanding of how real systems (like gases) behave. So let's explore the behavior of this system by laying out 20 coins (I suggest laying them out in four rows with five coins in each row). We will start with all 20 coins heads up and then we will choose which coin to turn over by rolling a 20-sided die. Flip over the coin that corresponds to the result of the die roll (the groups of five should make it easy to find the correct coin). 23. Continue the experiment until you have completed 80 coin flips. Keep track of the results in a table or a spreadsheet (if available). You should have columns for the number of flips (starting at 0), the number of heads (initially 20), and the number of tails (initially 0). 24. Create a plot of the number of heads after each flip. If your data is in a spreadsheet, use an X-Y plot with the points connected by a line. Make sure to label your axes and give the plot a title. 25. Describe what happens to the number of heads in your sequence of coins as you flip more and more coins.