Pass the Parcel

You might like to look at the problem Markov Matrices before attempting this problem.

Four children, $A$, $B$, $C$ and $D$, are playing a version of the game "pass the parcel". They stand in a circle, so that $ABCDA$ is the clockwise order. Each time a whistle is blown, the child holding the parcel is supposed to pass the parcel immediately exactly one place clockwise.

In fact each child, independently of any other past event, passes the parcel clockwise with probability $\frac{1}{4}$, passes it anticlockwise with probability $\frac{1}{4}$ and fails to pass it at all with probability $\frac{1}{2}$.

1. Write down the transition matrix, ${\bf M}$, for this situation.
    
2. Calculate ${\bf M}^2$, ${\bf M}^3$ and ${\bf M}^4$ (you can use a calculator or online matrix multiplier to help you!).
    
3. Suggest a general form for ${\bf M}^n$.
    
4. If the game starts with child $A$ holding the parcel, work out the probabilities that after $n$ whistle blasts $A$ is holding the parcel.  Find also the probabilities that $B$, $C$ and $D$ are holding the parcel.
    
5. What can you say about the probabilities that each child is holding the parcel after a long time (so as $n \to \infty)$?

You may find these Matrix Power calculators useful:

• Matrix Power Calculator (decimal version)
• Matrix Power Calculator (fraction version)

There are more matrix problems in this feature.

Based on STEP Mathematics 2, 2018, Q13. Question reproduced by kind permission of Cambridge Assessment Group Archives. The question remains Copyright University of Cambridge Local Examinations Syndicate ("UCLES"), All rights reserved.