## Parcel Passing Game Example

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 |

\[T= \left( \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 1/3 & 0 & 1/3 & 1/3 \\ 0 & 1/2 & 0 & 1/2 \\ 1/3 & 0 & 2/3 & 0 \end{array} \right)\]

.\[T^2\]

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.