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PaperclipPerfector  3mo ago (text post)   2298 thread views

Friday Fun Thread for September 6, 2024

Be advised: this thread is not for serious in-depth discussion of weighty topics (we have a link for that), this thread is not for anything Culture War related. This thread is for Fun. You got jokes? Share 'em. You got silly questions? Ask 'em.

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Two intuitive ways to formulate this come to mind.

First, law of conditional probability. P(A | B) * P(B) = P(A and B), or alternatively, P(A | B) = P(A and B) / P(B). P(2nd Coin Gold | 1st Coin Gold) = P(Both Coins Gold) / P(1st Coin Gold). P(1st Coin Gold) is 50% or 1/2, the probability of selecting the box with both gold coins (1/3) plus the probability of selecting the mixed coin box times one half (1/3 * 1/2 = 1/6). Probability both coins are gold is 1/3, as it can only occur if you pick the one box of three with two gold coins. 1/3 / (1/2) = 2/3.

Second, just thinking of all the possible outcomes before any coins are picked. Let's arbitrarily designate one coin in each box X, and the other Y. So boxes one to three contain coins Gold X and Gold Y, Gold X and Silver Y, and Silver X and Silver Y, respectively. There are only six possible outcomes for first and second pick, comma separated:

  1. Gold X, Gold Y
  2. Gold X, Silver Y
  3. Silver X, Silver Y
  4. Gold Y, Gold X
  5. Silver Y, Gold X
  6. Silver Y, Silver X

If you picked gold first, you know you can't be in outcomes 3, 5, or 6. Out of outcomes 1, 2, and 4, two of those have gold as the second pick. Thus, 2/3 once again.

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