Can You Beat the Odds in This Probability Challenge?

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SUMMARY

The probability challenge involves two distinct secret numbers, X and Y, where X < Y. A fair coin flip determines which number is revealed: if Heads, Y is shown; if Tails, X is shown. The objective is to guess the coin flip outcome with a probability greater than 0.5 based solely on the revealed number. Participants discuss the necessity of a finite domain for X and Y to devise a successful strategy, indicating that without specified bounds, achieving a probability greater than 0.5 is unlikely.

PREREQUISITES
  • Understanding of basic probability concepts
  • Familiarity with discrete and continuous probability distributions
  • Knowledge of coin flip probability mechanics
  • Ability to analyze conditional probabilities
NEXT STEPS
  • Research strategies for conditional probability in finite domains
  • Explore the implications of known distributions on probability outcomes
  • Study methods for increasing guessing accuracy in probabilistic scenarios
  • Learn about Bayesian inference and its application in probability challenges
USEFUL FOR

This discussion is beneficial for students in introductory probability courses, mathematicians interested in probability theory, and anyone looking to enhance their problem-solving skills in probabilistic contexts.

JasonJo
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I know two distinct secret numbers: call them X and Y, and assume that X < Y, without loss of generality. You have no clue how I came up with them. They could be anything (positive, negative, rational, irrational, etc). They could come from any probability distribution (discrete or continuous). You have no idea. I flip a fair coin. If the coin shows Heads, I reveal to you the larger number, Y; if it shows Tails, I reveal to you the smaller number, X. You do not get to see the result of the coin flip. Your goal is to guess whether the coin was Heads or Tails, based only on your seeing the one number that I revealed to you. Obviously, if you just decide ``Heads'' is your guess, without taking into account the revealed number at all, then you are correct with probability 0.5. But your goal is to be able to be correct with probability {\em strictly greater} than 0.5. Devise a method to do this, and explain your solution

anyone know how to approach this? its obviously not for homework, I am in an intro to prob course.
 
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I don't think you stand a chance unless the domain of X and Y values is finite and specified beforehand.
 
thats what I am thinking too,

i'll let you guys know what the answer is probably today if you're interested (i'm going to his office hours).

cya!
 
Never bring a knife to a gun fight.

Carl
 

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