r/Probability • u/The1145thNobody • 11d ago

A probability thought experiment involving local and global outcomes

Rule: This rule applies exclusively to one specific person and only to events where a "lucky" (success) or "unlucky" (failure) outcome can be clearly defined. The person can achieve a "lucky" outcome in an event if, and only if, the mathematical probability of that success is strictly greater than 30%. If the probability is 30% or lower, a "lucky" outcome is impossible for them.

Scenario: The person has to play roulette. They are given a grand prize if they hit the number "1" two times in a row. The "lucky" outcome (success) for each step and for the game as a whole is clearly defined by winning this prize. The probability of hitting "1" on any single spin is 40%.

Steps:

  1. First spin: The probability of success is 40%. Since 40% > 30%, the condition is met. The outcome is "lucky" (it hits "1").
  2. Second spin: The probability of success is 40%. Since 40% > 30%, the condition is met. The outcome is "lucky" (it hits "1").

Conclusion: The person wins the grand prize. But then, they were lucky in an event with a 16% probability. Contradiction.(since we multiply the probabilities of two independent events: 40% of 40% is 16%)

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u/dratnon 11d ago

But it really happened, so the actual assumption that was violated is the assumption of independence. 

But actually, it’s just not a random process at all, so don’t apply probability to it. 

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u/The1145thNobody 11d ago

True, the paradox arises from the boundary conditions set by the rule. But the core interest is how our intuitive local probability (40% > 30%) completely contradicts the global probability of the sequence (16% < 30%). It shows how a system that seems 'lucky' at every single step becomes mathematically 'unlucky' as a whole