Flipping coins that gain and lose

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SUMMARY

The discussion centers on a probability problem involving a coin-flipping game where participants bet $100, earning $10 for heads and losing $10 for tails. The objective is to determine the probability of quitting with a net gain of $120 before losing $20. Participants explore the relationship to the Negative Binomial distribution, noting its limitations in this context due to the nature of losses. Simplifying the problem by considering pairs of coin tosses is suggested as a potential strategy for analysis.

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  • Understanding of basic probability concepts, including expected value and outcomes.
  • Familiarity with the Negative Binomial distribution and its applications.
  • Knowledge of coin-flipping games and their mathematical modeling.
  • Ability to analyze sequences of events and their implications on outcomes.
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  • Research the properties and applications of the Negative Binomial distribution in gambling scenarios.
  • Learn about Markov chains and their relevance in modeling stochastic processes like coin flips.
  • Explore simulations of coin-flipping games to visualize probabilities and outcomes.
  • Study strategies for optimizing betting games based on probability theory.
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Mathematicians, statisticians, game theorists, and anyone interested in probability theory and its applications in gambling scenarios.

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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?
 
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Welcome to PF!

Hi ProbProblems! Welcome to PF! :smile:
ProbProblems said:
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 :rolleyes: …

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. :wink:​
 

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