Easy math puzzles

Quote: Wizard

p.s. Why am I advised to capitalize Googleplex?
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I think "googolplex" is the number, and "Googleplex" is Alphabet City.

I think that the answer depends on how the location of the portal is chosen. I assume that either path is equally likely but the probability distribution of the distance along the correct path is important (and without a rigorous definition of this we can't calculate the expected distance traveled, so we can't minimize it).

The probability distribution can't be uniform because there can't be a uniform probability distribution over an unbounded set (you can have one over an infinite set, like all the real numbers between 0 and 1, but it needs to be bounded -- there's no uniform probability distribution over all the positive reals)

Quote: Wizard

An Evil Wizard transports you to a distant planet, where you are placed at the beginning of two different paths. As in the Led Zeppelin song, there are two paths you may go down and there's always time to go back and switch to the other path any time you wish as many times as you wish.

Somewhere, on one of the paths, is a portal taking you back home.

The path with the portal has a finite but unknown distance (for example, an inch or a Googleplex light years). The other is infinitely long.

What strategy should you take to minimize the expected walking time to find the portal?

p.s. Why am I advised to capitalize Googleplex?
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This problem has already been posted twice before and never answered. Even the “googleplex” term was used. Formerly it was framed as an evil wizard in an infinite “but not completely infinite” desert. Don’t waste your time on it