Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• Consider a room full of air molecules. Each molecule can be considered to be in one of two states: in the right half of the room or in the left half. The probability of each of these states is 1/2. If there are N distinguishable molecules in the room, then the total number of ways of arranging the molecules is 2N. (a) Argue that the number of ways of arranging the N distinguishable molecules with NL molecules on the left is

(b) If there are 100 molecules in the room, what is the ratio PL(0.5)/PL(0.6); that is, how much more likely is it that the molecules are evenly distributed compared to having 60 on the left and 40 on the right? (c) If there are 1025 molecules in the room, what is the ratio (PL = 0.500)/(PL = 0.501)? To compute the factorial of large numbers, use Stirling’s approximation: ln N! ≈ N ln N − N.