Jill has 20 dollars in her pocket. She goes to the Crown Casino in Melbourne and gambles. She would get 10 dollars if she won and would lose 10 dollars if she lost. She would win the game if she got 50 dollars or end if she went broke. She wins a game with probability $p$ and loses with probability $1-p$.
a) what is the probability that she wins the game in three turns
b)if the probability of the game ending is $q$ for a game to end. And if she ends the game in 5 turns then $q_5=1$ for n ∈ {1,2,3,4} prove that $q_n=pq_{n+1} +(1-p)q_{n-1}$ for $n$ ∈ $\{0,1,2,3,4,5\}$
c)if the difference between $q_{n+1}$, $q_n$ is $d_n$, then $d_{n-1} =\alpha d_n.$ what is the value of $α$.
I think the answer for first one is $p^3$. I have no clue for b and c parts. Any help is appreciated
For b n c part do I have to make use of the Random Walk strategy?