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So consider this question:

Suppose you are on the integral number line then if you toss a biased coin with probability "p" for heads and "q" for tails. You start at 0 and you toss a coin if its heads you move right and if its tails you move left and the line is infinite. If you move to the point "1" (i.e. you move right from 0 ) you lose the game. What's the probability that you do not lose the game?

My approach was: Since we have to not go to 1: Lets compute $1-prob(lose)$ for the $prob(win)$

$prob(lose)$ would be $p+qp^{2}+q^{2}p^{3}...$ because moving 1 step to right or moving 1 step left and 2 step right and so on

But the answer was in this format $prob(lose) = p+q \cdot prob(lose)^{2}$ I cant understand how it came any help is appreciated!

Robert D-B
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Razz
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  • If you have moved to $-1$ then, in order to lose you must first return to $0$ (probability $=p_{lose}$) and then from there you have to move to $1$ (probability $p_{lose}$ again). Note that this isn't enough to solve the problem in general. It's not even clear that $p_{lose}$ is well defined, and even if it is, the roots are $1,\frac p{1-p}$ and it isn't obvious which one is correct. – lulu Apr 28 '23 at 17:22
  • How is $p(lose)$ = probability of going to right i can't understand that we are moving to right right and we are not losing so doesn't it just come about the value of tails ?? @lulu – Razz Apr 28 '23 at 17:45
  • Since the number line is uniform, $p_{lose}$ is just the probability that at some point you will be one unit to your right. Thus, if you are at $-119$, to choose a random value, the probability that you will eventually find yourself at $-118$ is also $p_{lose}$. – lulu Apr 28 '23 at 17:47
  • I'm sorry but I can't satisfactorly understand this , this is my main question too it would be great if you formulate an answer regarding this because how is plose just one unit to the right then by logic wouldn't pwin be one unit to the left??? @lulu – Razz Apr 28 '23 at 17:53
  • Look at the duplicate... The answers given there are pretty complete. – lulu Apr 28 '23 at 18:38
  • Thank you now i understand @lulu – Razz Apr 29 '23 at 08:21

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