Tuesday, October 8, 2019

Probability Puzzle: Mutating Loaded Die


Take an (initially) fair six-sided die (i.e. $P(x)=\frac{1}{6}$ for $x=1,…,6$) and roll it repeatedly.



After each roll, the die becomes loaded for the next roll depending on the number $y$ that was just rolled according to the following system:


$$P(y)=\frac{1}{y}$$ $$P(x)=\frac{1 - P(y)}{5} \text{, for } x \ne y$$


i.e. the probability that you roll that number again in the next roll is $\frac{1}{y}$ and the remaining numbers are of equal probability.


What is the probability that you roll a $6$ on your $n$th roll?



NB: This is not a homework or contest question, just an idea I had on a boring bus ride. Bonus points for calculating the probability of rolling the number $x$ on the $n$th roll.


Answer



The transition matrix is given by $$\mathcal P = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ \tfrac{1}{10} & \tfrac{1}{2} & \tfrac{1}{10} & \tfrac{1}{10} & \tfrac{1}{10} & \tfrac{1}{10} \\ \tfrac{2}{15} & \tfrac{2}{15} & \tfrac{1}{3} & \tfrac{2}{15} & \tfrac{2}{15} & \tfrac{2}{15} \\ \tfrac{3}{20} & \tfrac{3}{20} & \tfrac{3}{20} & \tfrac{1}{4} & \tfrac{3}{20} & \tfrac{3}{20} \\ \tfrac{4}{25} & \tfrac{4}{25} & \tfrac{4}{25} & \tfrac{4}{25} & \tfrac{1}{5} & \tfrac{4}{25} \\ \tfrac{1}{6} & \tfrac{1}{6} & \tfrac{1}{6} & \tfrac{1}{6} & \tfrac{1}{6} & \tfrac{1}{6} \end{bmatrix}.$$ It is fairly easy to get numerical values for the probability distribution of being in state $6$ after $n$ steps, but a closed form solution appears difficult.


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