i stumbled upon this lecture notes regarding solving Gambler's Ruin problem "https://web.mit.edu/neboat/Public/6.042/randomwalks.pdf", and the lecturer concluded that
pn = (p/ 1-p)^m
after linear recurrence of
pn = pPn+1 + (1 − p)Pn−1
where "n" start amount of money , "m" is the amount of dollars you want to win . T = n+m.
In another source , it stated The probability of being at position m after N jumps is therefore given a bionomial distribution equation.
Which one is right ?!