The Joe Shmo Show

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Posers and Puzzles 20 Apr '21 14:41
  1. R
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    Luckily...you have finally made it on the Joe Shmo giveaway show!

    You have in front of you two identical treasure chests ( on the exterior ). I tell you that one chest contains 100 silver coins and the other contains 50 silver and 50 gold coins. You are then blindfolded and asked to pick a chest at random and draw a coin: Its a silver coin - its yours to keep, so you place it in your pocket. To your surprise I then give you a second chance! If you draw a gold coin you keep the entire contents of the chest; if not, you just get the coins you drew ( everyone walks away richer on the Shmo Show! )

    What is the ( what I'm naming ) relative likelihood 𝑅 you draw a gold coin if you switch chests vs. stay for the next draw?

    𝑅 = 𝑃( Gold | Switch ) / 𝑃( Gold | Stay)

    Please hide your response so everyone get a chance to play!
  2. R
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    20 Apr '21 19:32
    @joe-shmo said
    Luckily...you have finally made it on the Joe Shmo giveaway show!

    You have in front of you two identical treasure chests ( on the exterior ). I tell you that one chest contains 100 silver coins and the other contains 50 silver and 50 gold coins. You are then blindfolded and asked to pick a chest at random and draw a coin: Its a silver coin - its yours to keep, so you pla ...[text shortened]... 𝑃( Gold | Switch ) / 𝑃( Gold | Stay)

    Please hide your response so everyone get a chance to play!
    Whoever has the grievance, please speak up. I'm all ears.
  3. santa cruz, ca.
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    20 Apr '21 19:39
    @joe-shmo said
    Luckily...you have finally made it on the Joe Shmo giveaway show!

    You have in front of you two identical treasure chests ( on the exterior ). I tell you that one chest contains 100 silver coins and the other contains 50 silver and 50 gold coins. You are then blindfolded and asked to pick a chest at random and draw a coin: Its a silver coin - its yours to keep, so you pla ...[text shortened]... 𝑃( Gold | Switch ) / 𝑃( Gold | Stay)

    Please hide your response so everyone get a chance to play!
    is this like the monty hall problem?
  4. R
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    20 Apr '21 19:51
    @lemondrop said
    is this like the monty hall problem?
    Might be?
  5. R
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    21 Apr '21 13:102 edits
    A warmup to the Shmo Show:

    1) What is the probability you drew a silver coin on the first draw?

    2) What is the the probability you chose the chest with 50 gold coins on your first draw?

    3) What is the the probability you chose the chest with 50 gold coins on your first draw, given that you drew a silver coin?
  6. R
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    29 Apr '21 11:21
    Still open.
  7. Joined
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    29 Apr '21 17:452 edits
    @lemondrop said
    is this like the monty hall problem?
    Not really, because the trick (which most people who pose it gloss over - Marilyn Vos "Savant" (in)famously didn't even understand that she'd done so) in that one is that Monty Hall knows what's behind the curtains. So, he actively changes the odds after the fact.

    In this case, the odds aren't actively changed, but your knowledge about them is, and therefore the result is still counterintuitive.


    It is, I rather think Reveal Hidden Content
    1 in 6 if you stick, and 1/3 if you switch
    . That's because your odds of drawing a silver coin are (this part, at least, is obvious) 1 if you picked the all-silver chest, and 1/2 if you picked the mixed one.

    And therefore, since it is given that you did pick a silver one, Reveal Hidden Content
    by the canonical interpretation of such problems, which I do not really find realistic but accept within probability theory, the odds are 2/3 that you chose the all-silver chest, and 1/3 that you chose the mixed one; and only in the latter case do you have 1/2 chance of picking a gold one next if you stick, and only in the former if you switch
    .


    Hold on, that's not right. It's almost right, but it ignores that you put the first coin in your pocket. The first three-quarters of the solution remain the same, but the last quarter turns out to be that Reveal Hidden Content
    if you picked the mixed chest, you now have a chance of 50/99 of picking a gold coin if your second chest is the mixed one, not 1/2
    .

    So the real odds are Reveal Hidden Content
    1/3 * 50/99 = 50/297 for sticking, and 2/3 * 50/99 = 100/297 for switching
    . Which in both cases is rather close to what I said before, but in both cases your odds are just slightly better.
  8. R
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    29 Apr '21 18:391 edit
    @shallow-blue said
    Not really, because the trick (which most people who pose it gloss over - Marilyn Vos "Savant" (in)famously didn't even understand that she'd done so) in that one is that Monty Hall knows what's behind the curtains. So, he actively changes the odds after the fact.

    In this case, the odds aren't actively changed, but your knowledge about them is, and therefore th ...[text shortened]... cases is rather close to what I said before, but in both cases your odds are just slightly better.
    Real close.

    P ( Stay | Gold ) = 1/3*50/99

    P ( Switch | Gold ) = 2/3*50/100

    R = P ( Switch | Gold ) / P ( Stay | Gold ) ≈ 2

    "but in both cases your odds are just slightly better"

    I don't agree with this statement above though...

    The odds are significantly better if you switch. You go from 16.8% to 33.3%
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