Probable gaming

Ashwin has a fair die with sides labeled 1 to 6. He first rolls the die and records it on a piece of paper. Then, every second thereafter, he re-rolls the die. If he rolls a different value than his previous roll, he records the value and continues rolling. If he rolls the same value, he stops, does not record his final roll, and computes the average of his previously recorded rolls. Given that Vincent first rolled a $1$ , let $E$ be the expected value of his result. There exist rational numbers $r,s,t>0$ such that $E=r-s ln t$ and $t$ is not a perfect power. If $r+s+t$ = $\frac{m}{n}$

for relatively prime positive integers $m$ and $n$ Please help

#Combinatorics

Easy Math Editor

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Comments

Here I got something

Let $E_{n,k}$ denote the expected value of the average after $n$ rolls ending with the number $k$ , then we have $E_{n,1}=\frac{5b_{n-1}}{n\cdot 6^{n-1}}+\sum_{m=2}^6\left(\frac{n-1}{n}\right)\frac{E_{n-1,m}}{6}$ where $b_n$ is the number of sequences of length $n$ ending in a $2$ . It's not hard to find that $b_n=5b_{n-2}+4b_{n-1}$ , $b_1=0$ , $b_2=1$ , so $b_n=\frac{5(-1)^n+5^n}{30}$ . For $k\neq 1$ , we have $E_{n,k}=\frac{n-1}{6n}\left(\sum_{m=1,m\neq k}^6E_{n-1,k}\right)+\frac{k\cdot b_n}{n\cdot 6^{n-1}}$ now let $E_n$ denote the expected average after $n$ rolls regardless of the last number rolled, then we have $E_n=\sum_{m=1}^6E_{n,m}=\frac{5(n-1)}{6n}E_{n-1}+\frac{5^{n-1}}{n\cdot 6^{n-1}}+\frac{15b_n}{n\cdot 6^{n-1}}$ where $E_1=1$ . By induction we arrive at $E_n=\left(\frac{5}{6}\right)^{n-1}+\frac{15}{n\cdot 6^{n-1}}\sum_{m=2}^nb_m\cdot 5^{n-m}$ Since Ashwin must roll the same number again in order to end the game, the desired value is $\frac{1}{6}\sum_{n=1}^\infty E_n=1+15\sum_{n=1}^{\infty}\frac{\left(\frac{1}{6}\right)^2}{n}\sum_{m=2}^nb_m\cdot 5^{n-m}$ Plugging in the explicit form for $b_n$ and using the Taylor series for $\ln(1+x)$ , this simplifies to $\frac{7}{2}-\frac{5}{12}\ln 7 \implies r+s+t=\frac{131}{12} \implies 100m+n=\boxed{13112}$

Rajyawardhan Singh - 1 year, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$