Flipping coins that gain and lose

[ProbProblems]

Homework Statement

You enter a game with a $100 bet that involves flipping coins. The coin has probability p that heads will come up, probability (1-p) that tails will come up. Now, if you are rewarded $10 for every heads, but lose $10 for every tails. You decide to quit if you lose or gain $20, so walk away with $80 or $120. Since you don't know how many flips it will take, what is the probability that you will end up quiting on a net gain (walking away with $120)?

Homework Equations

The Attempt at a Solution

Immediately, I think it's probability p, but I'm having a hard time showing why. This is close to a Negative Binomial, but that is good for trials until r successes, and doesn't count for you losing. Any thoughts?

 

[tiny-tim]

Welcome to PF!

Hi ProbProblems! Welcome to PF!

You enter a game with a $100 bet that involves flipping coins. The coin has probability p that heads will come up, probability (1-p) that tails will come up. Now, if you are rewarded $10 for every heads, but lose $10 for every tails.

hmm …

let's simplify this by ignoring all the odd number of tosses …

on every even toss, either you've stopped, or you're back to the start …

so toss two coins at a time. ​