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In Wikipedia, if Player 1 has $n_1$ pennies, and Player 2 has $n_2$ pennies, and they play the game where they flip a fair coin, if it comes up heads, Player 1 gives a penny to Player 2, and if its tails, he receives a penny from the other player, and they play until someone has no more pennies left, the probability of Player 1 winning is $\frac{n_1}{n_1 + n_2}$.

I confirmed this by doing the whole process for Gambler's ruin, but I am not sure how it's obvious intuitively. It's written in Wikipedia as if it's something obvious, but I don't see how you can derive this without essentially doing the computation for showing the Gambler's Ruin problem.

It would be great to understand the intuition.

Thank you!

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    $\begingroup$ It's certainly a plausible answer - you need something symmetric when interchanging the players, and why shouldn't my chance of winning be simply the fraction of the total coins I have - but I don't think the Wikipedia article is written as if it's obvious. It's just stating a fact. $\endgroup$ Mar 28, 2022 at 8:35

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Player $1$ has an equal probability of $0.5$ to either move up by one penny, or down by one penny, $P(n) = 0.5[P(n+1) +P(n-1)$],
or $P(n+1) - P(n) = P(n) - P(n-1)$,
which means that the relationship is linear

From the point we are at the start, ruin will occur after $n_1$ steps in one direction and $n_2$ steps in the other in an ambit of $(n_1+n_2)$ steps

And intuitively, it is obvious that the more coins $A$ has, the more probable that he wins, thus P(A wins) = $\dfrac{n_1}{n_1+n_2}$

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