Parcel Passing Game Example

Four people A, B, C and D play pass the parcel according to the following rule.
A always passes to B.
Bob is equally likely to pass to anyone.
C passes only to A or B with equal probability.
D passes to C twice as often as they do to A and never passes to B.
What is the probability that the parcel will pass from B to C in two passes?
What is the probability that the parcel will pass from C to A in three passes?
The table of probabilities is:
 Passing Player\Receiving Player A B C D A 0 0 0 1 B 1/3 0 1/3 1/3 C 0 1/2 0 1/2 D 1/3 0 2/3 0
The transition matrix is
$T= \left( \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 1/3 & 0 & 1/3 & 1/3 \\ 0 & 1/2 & 0 & 1/2 \\ 1/3 & 0 & 2/3 & 0 \end{array} \right)$
.
$T^2$
tells us the probabilities of the parcel starting at one player and passing to another after two passes.
$T^2= \left( \begin{array}{cccc} 1/3 & 0 & 2/3 & 0 \\ 1/9 & 1/6 & 2/9 & 1/2 \\ 1/3 & 0 & 1/2 & 1/6 \\ 0 & 1/3 & 0 & 2/3 \end{array} \right)$

The probabilities of passes between any two players in two passes is given by
$T^2$
, and the probability of passing from B to C in two steps is the entry in row 2, column 3, and is equal to 2/9.
$T^3$
tells us the probabilities of the parcel starting at one player and passing to another after three passes.
$T^3= \left( \begin{array}{cccc} 0 & 1/3 & 0 & 2/3 \\ 2/9 & 1/9 & 7/18 & 5/18 \\ 1/18 & 1/4 & 1/9 & 7/12 \\ 1/3 & 0 & 5/9 & 1/9 \end{array} \right)$

The probabilities of passes between any two players in three passes is given by
$T^3$
, and the probability of passing from C to A in two steps is the entry in row 3, column 1, and is equal to 1/18.