Largest and smallest possible values of a probability question

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Discussion Overview

The discussion revolves around a probability problem involving the determination of the largest and smallest possible values of P({2}) based on given probabilities of certain sets of integers. The scope includes mathematical reasoning and exploratory analysis of probability distributions.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant presents a probability problem involving the random selection of a positive integer and known probabilities for specific sets.
  • Another participant calculates that P({2,3,4,5}) = 0.2 and suggests that P({4,5}) is less than or equal to this value.
  • A later reply indicates that P({2}) is suspected to be between 0 and 0.2, noting the lack of restrictions on P({3}) or P({6}).
  • Participants express uncertainty about whether they are missing additional constraints that could affect the bounds on P({2}).

Areas of Agreement / Disagreement

Participants generally agree that P({2}) is likely between 0 and 0.2, but there is uncertainty regarding any additional constraints that might apply.

Contextual Notes

The discussion does not resolve the potential dependencies on the values of P({3}) and P({6}), which could influence the bounds on P({2}).

PhysicsMathGuy
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Hello,
So I've been trying to solve this problem for a while and I can't get my head around it.
"Suppose we choose a positive integer at random, according to some unknown distribution. Suppose we know P({1,2,3,4,5}) = 0.3, P({4,5,6}) = 0.4 and P({1}) = 0.1"

All I've managed to get so far is that P({2,3,4,5}) = P({1,2,3,4,5}) - P({1}) = 0.2
I'm sure that's not a good upper bound because there's probably a way the use the fact that the union of {1,2,3,4,5} and {4,5,6} is {4,5}.

Thanks in advance for your help!
 
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Just saw that this section is not for this. Sorry, my bad!
 
P({4,5}) ≤ P({2,3,4,5}) =0.2, P({6}) = P({4,5,6}) - P({4,5}) ≥ 0.2. What are you trying to find?
 
Thanks a lot for your reply! I'm trying to find the largest and smallest possible values of P({2}).
 
Off hand P({2}) is between 0 and 0.2. I suspect this is the best you can do since there is no restriction on P({3}) or P({6}).
 
Thanks a lot! That's what I thought, but I wasn't sure and thought I was missing something. Thanks a lot!
 

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