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You start off with a \$100 pool. Every second, your pool doubles in value with probability $p$ or gets cut in half with probability $1-p.$ If you drop below 98.76543210ยข at any point, you lose and the host of the game show laughs at you.

Furthermore, you stubbornly refuse to stop playing. What's the probability that you eventually lose? Note: Assume that you and the host are both immortal, and that thermodynamic restrictions do not prevent gameplay in the longue duree.

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Meta: This is in effect a standard question in the theory of random walks. Does that make it off-topic as a textbook problem? I don't think so. The relevant meta-question is this one whose excellent and highest-voted answer gives this closely related question as an example of something that's a "math puzzle" rather than a "math problem".

OK then. Assuming that there's no weird business with rounding of fractional cents, this question is equivalent to the following: You start with a number $x=7$. You repeatedly add 1 (with probability $p$) or subtract 1 (with probability $1-p$). What's the probability that you eventually reach $x=0$? (Why 7? Because 6 halvings of \$100 take you to \$1.5625 (above the threshold in the question) and 7 halvings take you to \$0.78125 (below the threshold).)

And the answer, which you can find in any suitable textbook, is

The probability is 1 if $p\leq\frac12$; otherwise it's $\bigl(\frac{1-p}{p}\bigr)^7$.

If you don't want to go out and find a suitable textbook, you can also deduce it from the accepted answer to that question I linked above: see that formula for $c_1,c_2$ and the way in which the answer to the original question is derived from it, and then put $a=-7$ instead of $a=-1$.

Meta again: Does that mean that this should have been closed as a duplicate? I think it's just different enough not to be: you can figure out the answer to this one from the answer to that one, but you can't just see it there.

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