The Ehrenfest model for heat exchange can be described as follows: Two urns, A and B, contain a total of 2N balls. At every step, one ball is equally likely to be drawn from among all 2N balls. It is then put into the other urn. Let Xn be the number of balls in urn A immediately after the nth step. Goldenberg (1988) uses this chain to represent futures and security prices, which tend to return to an equilibrium N. The greater the distance between the current price and N, the greater the probability that it moves toward N rather than away from it. Required:
Find the transition matrix for this process.

Martha has a deck of cards. she has lost some of the cards, and now the deck only contains nine spades, eleven diamonds, eight clubs, and twelve hearts. martha predicts that whenever she draws a card from the deck without looking, she will draw a club one-fifth of the time. which activity would best allow martha to test her prediction? a. randomly draw a card from the box and see if it is a club. b. randomly draw a card. then, continue to draw another card until all eight clubs are drawn. c. randomly draw and replace a card 120 times. then, observe how close to 30 times a club is drawn. d. randomly draw and replace a card 100 times. then, observe how close to 20 times a club is drawn.