{VERSION 2 3 "APPLE_68K_MAC" "2.3" }
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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 13 "Markov Chains" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 189 "The cell below
 defines the matrix giving the probabilities of moving from one state \+
to another and the initial probability vector.  Then it uses matrix mu
ltiplication to study this model.  " }{TEXT 256 16 "Evaluate it now." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1139 "A := array(1..12, 1..12):\n\nA := [[0, 0, 0, 0, 0, 0, 0, 0, \+
0, 0, 0, 0],\n      [6/36, 1, 0, 2/36, 3/36, 4/36, 5/36, 5/36, 4/36, 3
/36, 2/36, 1/36],\n      [1/36, 0, 1, 6/36, 6/36, 6/36, 6/36, 6/36, 6/
36, 6/36, 6/36, 6/36],\n      [2/36, 0, 0, 28/36,   0,    0,    0,    \+
0,    0,    0,    0,    0],\n      [3/36, 0, 0,     0, 27/36,  0,    0
,    0,    0,    0,    0,    0],\n      [4/36, 0, 0,     0,   0, 26/36
,   0,    0,    0,    0,    0,    0],\n      [5/36, 0, 0,     0,   0, \+
   0, 25/36,   0,    0,    0,    0,    0],\n      [5/36, 0, 0,     0, \+
  0,    0,    0, 25/36,   0,    0,    0,    0],\n      [4/36, 0, 0,   \+
  0,   0,    0,    0,    0, 26/36,   0,    0,    0],\n      [3/36, 0, \+
0,     0,   0,    0,    0,    0,    0, 27/36,   0,    0],\n      [2/36
, 0, 0,     0,   0,    0,    0,    0,    0,    0, 28/36,   0],\n      \+
[1/36, 0, 0,     0,   0,    0,    0,    0,    0,    0,    0, 29/36]]:
\n\nP0 := [[1.0], [0.0], [0.0], [0.0], [0.0], [0.0], \n       [0.0], [
0.0], [0.0], [0.0], [0.0], [0.0]]:\n\nP := proc(n) option remember;\n \+
    if n = 0 then P0\n        else evalm(A &* P(n - 1));\n     fi\n   \+
  end:\n\nP(1);\nP(2);\nP(3);\nP(4);\nP(10);\nP(20);\nP(40);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 374 "N
otice that as the game is played a player is more likely to find himse
lf or herself in states 2 or 3 -- the stwo states corresponding to com
pleted games.  After 40 throws of the dice a player will have won (sta
te 2) with probability 46.91% and will have lost with probability 53.0
9%.\n\nThe cell below shows how these computations can be done without
 matrix multiplication.  " }{TEXT 257 15 "Evaluate it now" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 275 "P0 := [1.0, 0.0, 0.0, 0.0, 0.0, 0.0, \n       0.0, 0
.0, 0.0, 0.0, 0.0, 0.0]:\n\nP := proc(n) option remember;\n     local \+
i, j:\n     if n = 0 then P0\n        else [seq(sum(A[i, j] * P(n - 1)
[j], j=1..12), i=1..12)];\n     fi\n     end:\n\nP(1);\nP(2);\nP(3);\n
P(4);\nP(10);\nP(20);\nP(40);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{MARK "0 2 1" 16 }{VIEWOPTS 1 1 0 1 1 1803 }