Black and white marbles probability question

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Discussion Overview

The discussion revolves around the probabilities associated with selecting marbles from a bag containing black and white marbles. Participants explore the implications of adding a black marble to a bag initially containing two marbles, examining how this affects the probability of selecting a black marble. The conversation includes theoretical reasoning and challenges regarding the interpretation of probabilities.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant presents initial probabilities for selecting marbles from a two-marble bag and calculates the probability of selecting a black marble after adding a third black marble.
  • Another participant challenges the conclusion that a 2/3 probability of selecting a black marble implies a specific configuration of the initial two marbles, suggesting alternative configurations could also yield the same probability.
  • Several participants clarify their understanding of the selection process and the implications of the probabilities discussed.
  • One participant suggests that enumerating the possible scenarios would clarify the situation, indicating a preference for a more systematic approach to the problem.
  • There is a recognition of the potential flaw in the reasoning regarding the exclusivity of the probability statement made by one participant.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the calculated probabilities, with some agreeing on the need for further clarification while others challenge the interpretations presented. The discussion remains unresolved regarding the correct understanding of the probabilities involved.

Contextual Notes

Participants note the complexity of the problem and the potential for misunderstanding the implications of the probabilities. There is an acknowledgment of the need to consider multiple configurations of the marbles that could lead to the same probability outcomes.

bahamagreen
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TL;DR
Bag with two marbles, each p=.5 of being black or white. After adding a third black marble, what is p of selecting a black marble from the bag? I get 2/3 which impossibly reveals the two marbles' colors. Not finding the flaw...
Two-marble probabilities
BB 1/4
BW 1/2
WW 1/4

Three-marble probabilities of selecting a black
B BB 1
B BW 2/3
B WW 1/3

Probability of selecting a black marble if we add a black marble to the two marble bag
(1 x 1/4) + (2/3 x 1/2) + (1/3 x 1/4) = 2/3

Three marbles with 2/3 probability of selecting black must mean two black and one white. If I remove (mentally) the black marble I added, I impossibly "know" the colors of the initial randomly selected two-marble bag (although I'm only right half the time).
 
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bahamagreen said:
Summary:: Bag with two marbles, each p=.5 of being black or white. After adding a third black marble, what is p of selecting a black marble from the bag? I get 2/3 which impossibly reveals the two marbles' colors. Not finding the flaw...

Three marbles with 2/3 probability of selecting black must mean two black and one white.
This is not correct. 2/3 probability of selecting black could also mean a 1/4 probability of one black and two white plus a 1/2 probability two black and one white plus a 1/4 probability of three black.
 
I mean selecting one marble and it being black...
 
Pretty trivial to just jot down the very small number of possible situations and it will be obvious what the answer is.
 
bahamagreen said:
I mean selecting one marble and it being black...
Yes, understood
 
Dale said:
Yes, understood
OK, thanks, been suspecting the "must" in that statement was too exclusive.
 

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