In [1]:
import sympy
In [2]:
oneby6=sympy.Rational(1,6)
In [3]:
Q = sympy.Matrix([[0 , oneby6, oneby6, oneby6, oneby6, oneby6, oneby6, 0 ],
                  [0 , 0,  oneby6, oneby6, oneby6, oneby6, oneby6, oneby6 ],
                   [0 , 0, 0, oneby6, oneby6, oneby6, oneby6, oneby6],
                 [0 , 0,0, 0, oneby6, oneby6, oneby6, oneby6 ],
                 [0 , 0,0,0,0, oneby6, oneby6, oneby6 ],
                 [0 , 0, 0,0,0,0, oneby6, oneby6 ],
                 [0 , 0, 0,0,0,0,0, oneby6],
                 [0 , 0,0,0,0,0,0,0]])
In [4]:
Q
Out[4]:
$\displaystyle \left[\begin{matrix}0 & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & 0\\0 & 0 & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6}\\0 & 0 & 0 & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6}\\0 & 0 & 0 & 0 & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6}\\0 & 0 & 0 & 0 & 0 & \frac{1}{6} & \frac{1}{6} & \frac{1}{6}\\0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{6} & \frac{1}{6}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{6}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{matrix}\right]$
In [5]:
Id8=sympy.eye(8)
In [7]:
IminusQ = Id8-Q
In [8]:
IminusQ
Out[8]:
$\displaystyle \left[\begin{matrix}1 & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6} & 0\\0 & 1 & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6}\\0 & 0 & 1 & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6}\\0 & 0 & 0 & 1 & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6}\\0 & 0 & 0 & 0 & 1 & - \frac{1}{6} & - \frac{1}{6} & - \frac{1}{6}\\0 & 0 & 0 & 0 & 0 & 1 & - \frac{1}{6} & - \frac{1}{6}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{1}{6}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{matrix}\right]$
In [9]:
inverseIminusQ= IminusQ.inv()
In [10]:
inverseIminusQ
Out[10]:
$\displaystyle \left[\begin{matrix}1 & \frac{1}{6} & \frac{7}{36} & \frac{49}{216} & \frac{343}{1296} & \frac{2401}{7776} & \frac{16807}{46656} & \frac{70993}{279936}\\0 & 1 & \frac{1}{6} & \frac{7}{36} & \frac{49}{216} & \frac{343}{1296} & \frac{2401}{7776} & \frac{16807}{46656}\\0 & 0 & 1 & \frac{1}{6} & \frac{7}{36} & \frac{49}{216} & \frac{343}{1296} & \frac{2401}{7776}\\0 & 0 & 0 & 1 & \frac{1}{6} & \frac{7}{36} & \frac{49}{216} & \frac{343}{1296}\\0 & 0 & 0 & 0 & 1 & \frac{1}{6} & \frac{7}{36} & \frac{49}{216}\\0 & 0 & 0 & 0 & 0 & 1 & \frac{1}{6} & \frac{7}{36}\\0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{1}{6}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{matrix}\right]$
In [11]:
rowOne=sympy.ones(8,1)
rowOne
Out[11]:
$\displaystyle \left[\begin{matrix}1\\1\\1\\1\\1\\1\\1\\1\end{matrix}\right]$
In [12]:
expectedUntilAbsorption = inverseIminusQ*rowOne
expectedUntilAbsorption
Out[12]:
$\displaystyle \left[\begin{matrix}\frac{776887}{279936}\\\frac{117649}{46656}\\\frac{16807}{7776}\\\frac{2401}{1296}\\\frac{343}{216}\\\frac{49}{36}\\\frac{7}{6}\\1\end{matrix}\right]$
In [13]:
776887/279936
Out[13]:
2.7752307670324647
In [ ]: