Ejemplo 2 matriz de transición

Suponga que en la industria una empresa tiene 5 proveedores, de los cuales, la probabilidad de que cambie de proveedor después de escoger alguno es la siguiente:


\left( \begin{matrix} 0.2 & 0.2 & 0.3 & 0.3 & 0.2 \\ 0.3 & 0.3 & 0.1 & 0.1 & 0.2 \\ 0.6 & 0.1 & 0.1 & 0.1 & 0.1 \\ 0.4 & 0.1 & 0.1 & 0.3 & 0.1 \\ 0.1 & 0.2 & 0.3 & 0.3 & 0.1 \\ \end{matrix} \right)=

¿Cuál es la probabilidad de que después de escoger el proveedor 1, en la segunda compra escoja el proveedor dos?

Solución


\text{C}=\text{A}\cdot \text{B}=\left( \begin{matrix} 0.2 & 0.2 & 0.3 & 0.3 & 0.2 \\ 0.3 & 0.3 & 0.1 & 0.1 & 0.2 \\ 0.6 & 0.1 & 0.1 & 0.1 & 0.1 \\ 0.4 & 0.1 & 0.1 & 0.3 & 0.1 \\ 0.1 & 0.2 & 0.3 & 0.3 & 0.1 \\ \end{matrix} \right)\cdot \left( \begin{matrix} 0.2 & 0.2 & 0.3 & 0.3 & 0.2 \\ 0.3 & 0.3 & 0.1 & 0.1 & 0.2 \\ 0.6 & 0.1 & 0.1 & 0.1 & 0.1 \\ 0.4 & 0.1 & 0.1 & 0.3 & 0.1 \\ 0.1 & 0.2 & 0.3 & 0.3 & 0.1 \\ \end{matrix} \right)=

\left( \begin{matrix} 0.42 & 0.2 & 0.2 & 0.26 & 0.16 \\ 0.27 & 0.21 & 0.2 & 0.22 & 0.16 \\ 0.26 & 0.19 & 0.24 & 0.26 & 0.17 \\ 0.3 & 0.17 & 0.2 & 0.26 & 0.15 \\ 0.39 & 0.16 & 0.14 & 0.2 & 0.13 \\ \end{matrix} \right)

Los componentes de la matriz C se calculan del modo siguiente:

C1,1=A1,1·B1,1+A1,2·B2,1+A1,3·B3,1+A1,4·B4,1+A1,5·B5,1=

= (0.2) · (0.2) + (0.2) · (0.3) + (0.3) · (0.6) + (0.3) · (0.4) + (0.2) · (0.1) = (0.04) + (0.06) + (0.18) + (0.12) + (0.02) = 0.42


C1,2=A1,1·B1,2+A1,2·B2,2+A1,3·B3,2+A1,4·B4,2+A1,5·B5,2=

= (0.2) · (0.2) + (0.2) · (0.3) + (0.3) · (0.1) + (0.3) · (0.1) + (0.2) · (0.2) = (0.04) + (0.06) + (0.03) + (0.03) + (0.04) = 0.2


C1,3=A1,1·B1,3+A1,2·B2,3+A1,3·B3,3+A1,4·B4,3+A1,5·B5,3=

= (0.2) · (0.3) + (0.2) · (0.1) + (0.3) · (0.1) + (0.3) · (0.1) + (0.2) · (0.3) = (0.06) + (0.02) + (0.03) + (0.03) + (0.06) = 0.2


C1,4=A1,1·B1,4+A1,2·B2,4+A1,3·B3,4+A1,4·B4,4+A1,5·B5,4=

= (0.2) · (0.3) + (0.2) · (0.1) + (0.3) · (0.1) + (0.3) · (0.3) + (0.2) · (0.3) = (0.06) + (0.02) + (0.03) + (0.09) + (0.06) = 0.26


C1,5=A1,1·B1,5+A1,2·B2,5+A1,3·B3,5+A1,4·B4,5+A1,5·B5,5=

= (0.2) · (0.2) + (0.2) · (0.2) + (0.3) · (0.1) + (0.3) · (0.1) + (0.2) · (0.1) = (0.04) + (0.04) + (0.03) + (0.03) + (0.02) = 0.16


C2,1=A2,1·B1,1+A2,2·B2,1+A2,3·B3,1+A2,4·B4,1+A2,5·B5,1=

= (0.3) · (0.2) + (0.3) · (0.3) + (0.1) · (0.6) + (0.1) · (0.4) + (0.2) · (0.1) = (0.06) + (0.09) + (0.06) + (0.04) + (0.02) = 0.27


C2,2=A2,1·B1,2+A2,2·B2,2+A2,2·B3,2+A2,4·B4,2+A2,5·B5,2=

= (0.3) · (0.2) + (0.3) · (0.3) + (0.1) · (0.1) + (0.1) · (0.1) + (0.2) · (0.2) = (0.06) + (0.09) + (0.01) + (0.01) + (0.04) = 0.21


C2,3=A2,1·B1,3+A2,2·B2,3+A2,3·B3,3+A2,4·B4,3+A2,5·B5,3=

= (0.3) · (0.3) + (0.3) · (0.1) + (0.1) · (0.1) + (0.1) · (0.1) + (0.2) · (0.3) = (0.09) + (0.03) + (0.01) + (0.01) + (0.06) = 0.2


C2,4=A2,1·B1,4+A2,2·B2,4+A2,3·B3,4+A2,4·B4,4+A2,5·B5,4=

= (0.3) · (0.3) + (0.3) · (0.1) + (0.1) · (0.1) + (0.1) · (0.3) + (0.2) · (0.3) = (0.09) + (0.03) + (0.01) + (0.03) + (0.06) = 0.22


C2,5=A2,1·B1,5+A2,2·B2,5+A2,3·B3,5+A2,4·B4,5+A2,5·B5,5=

= (0.3) · (0.2) + (0.3) · (0.2) + (0.1) · (0.1) + (0.1) · (0.1) + (0.2) · (0.1) = (0.06) + (0.06) + (0.01) + (0.01) + (0.02) = 0.16


C3,1=A3,1·B1,1+A3,2·B2,1+A3,3·B3,1+A3,4·B4,1+A3,5·B5,1=

= (0.6) · (0.2) + (0.1) · (0.3) + (0.1) · (0.6) + (0.1) · (0.4) + (0.1) · (0.1) = (0.12) + (0.03) + (0.06) + (0.04) + (0.01) = 0.26


C3,2=A3,1·B1,2+A3,2·B2,2+A3,3·B3,2+A3,4·B4,2+A3,5·B5,2=

= (0.6) · (0.2) + (0.1) · (0.3) + (0.1) · (0.1) + (0.1) · (0.1) + (0.1) · (0.2) = (0.12) + (0.03) + (0.01) + (0.01) + (0.02) = 0.19


C3,3=A3,1·B1,3+A3,2·B2,3+A3,3·B3,3+A3,4·B4,3+A3,5·B5,3=

= (0.6) · (0.3) + (0.1) · (0.1) + (0.1) · (0.1) + (0.1) · (0.1) + (0.1) · (0.3) = (0.18) + (0.01) + (0.01) + (0.01) + (0.03) = 0.24


C3,4=A3,1·B1,4+A3,2·B2,4+A3,3·B3,4+A3,4·B4,4+A3,5·B5,4=

= (0.6) · (0.3) + (0.1) · (0.1) + (0.1) · (0.1) + (0.1) · (0.3) + (0.1) · (0.3) = (0.18) + (0.01) + (0.01) + (0.03) + (0.03) = 0.26


C3,5=A3,1·B1,5+A3,2·B2,5+A3,3·B3,5+A3,4·B4,5+A3,5·B5,5=

= (0.6) · (0.2) + (0.1) · (0.2) + (0.1) · (0.1) + (0.1) · (0.1) + (0.1) · (0.1) = (0.12) + (0.02) + (0.01) + (0.01) + (0.01) = 0.17


C4,1=A4,1·B1,1+A4,2·B2,1+A4,3·B3,1+A4,4·B4,1+A4,5·B5,1=

= (0.4) · (0.2) + (0.1) · (0.3) + (0.1) · (0.6) + (0.3) · (0.4) + (0.1) · (0.1) = (0.08) + (0.03) + (0.06) + (0.12) + (0.01) = 0.3


C4,2=A4,1·B1,2+A4,2·B2,2+A4,3·B3,2+A4,4·B4,2+A4,5·B5,2=

= (0.4) · (0.2) + (0.1) · (0.3) + (0.1) · (0.1) + (0.3) · (0.1) + (0.1) · (0.2) = (0.08) + (0.03) + (0.01) + (0.03) + (0.02) = 0.17


C4,3=A4,1·B1,3+A4,2·B2,3+A4,3·B3,3+A4,4·B4,3+A4,5·B5,3=

= (0.4) · (0.3) + (0.1) · (0.1) + (0.1) · (0.1) + (0.3) · (0.1) + (0.1) · (0.3) = (0.12) + (0.01) + (0.01) + (0.03) + (0.03) = 0.2


C4,4=A4,1·B1,4+A4,2·B2,4+A4,3·B3,4+A4,4·B4,4+A4,5·B5,4=

= (0.4) · (0.3) + (0.1) · (0.1) + (0.1) · (0.1) + (0.3) · (0.3) + (0.1) · (0.3) = (0.12) + (0.01) + (0.01) + (0.09) + (0.03) = 0.26


C4,5=A4,1·B1,5+A4,2·B2,5+A4,3·B3,5+A4,4·B4,5+A4,5·B5,5=

= (0.4) · (0.2) + (0.1) · (0.2) + (0.1) · (0.1) + (0.3) · (0.1) + (0.1) · (0.1) = (0.08) + (0.02) + (0.01) + (0.03) + (0.01) = 0.15


C5,1=A5,1·B1,1+A5,2·B2,1+A5,3·B3,1+A5,4·B4,1+A5,5·B5,1=

= (0.1) · (0.2) + (0.2) · (0.3) + (0.3) · (0.6) + (0.3) · (0.4) + (0.1) · (0.1) = (0.02) + (0.06) + (0.18) + (0.12) + (0.01) = 0.39


C5,2=A5,1·B1,2+A5,2·B2,2+A5,3·B3,2+A5,4·B4,2+A5,5·B5,2=

= (0.1) · (0.2) + (0.2) · (0.3) + (0.3) · (0.1) + (0.3) · (0.1) + (0.1) · (0.2) = (0.02) + (0.06) + (0.03) + (0.03) + (0.02) = 0.16


C5,3=A5,1·B1,3+A5,2·B2,3+A5,3·B3,3+A5,4·B4,3+A5,5·B5,3=

= (0.1) · (0.3) + (0.2) · (0.1) + (0.3) · (0.1) + (0.3) · (0.1) + (0.1) · (0.3) = (0.03) + (0.02) + (0.03) + (0.03) + (0.03) = 0.14


C5,4=A5,1·B1,4+A5,2·B2,4+A5,3·B3,4+A5,4·B4,4+A5,5·B5,4=

= (0.1) · (0.3) + (0.2) · (0.1) + (0.3) · (0.1) + (0.3) · (0.3) + (0.1) · (0.3) = (0.03) + (0.02) + (0.03) + (0.09) + (0.03) = 0.2


C5,5=A5,1·B1,5+A5,2·B2,5+A5,3·B3,5+A5,4·B4,5+A5,5·B5,5=

= (0.1) · (0.2) + (0.2) · (0.2) + (0.3) · (0.1) + (0.3) · (0.1) + (0.1) · (0.1) = (0.02) + (0.04) + (0.03) + (0.03) + (0.01) = 0.13