re: Solve this OT

PearlJam, use ell's example then.

GGG, GGS, GSS, SSS

You've chosen a gold ball. How many possible outcomes are there for your next pull?

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Because 1 ball remains in the box. It can only be gold or silver.

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If you had a box with 3 gold balls, and another box with 1 gold ball and two silvers, what are the odds of pulling a second gold ball from the same box if your first pull resulted in gold.

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75%!!

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correct!

But I want Pearl Jam to answer why

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Once you choose a gold ball, there are only 2 outcomes left. But you're twice as likely to choose said gold ball.

Right?

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It's been 6 pages of semantics.

There are 3 possible pairs involving gold balls.

G1, G2
G2, G1
G3, S1

Once you've pulled a gold ball, there are three possible balls that still remain - G1, G2, or S1. There are two possible colors that still remain.

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If you understand why the answer is 2/3 then solve that for me. Show your work.

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There's a 67% chance you chose a gold ball from teh double gold box.

When you choose that gold ball, the next will be gold 100% of the time. When you choose from the other box, it'll be silver 100% of the time.

So the 67% of the time you choose the double gold box is essentially the answer to how often the next ball will be gold.

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GGG, GGS, GSS, SSS

You've chosen a gold ball. How many possible outcomes are there for your next pull?

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Here's my logic:

I know I'm going to pull a gold so, 50% chance for Box 1, 33% chance for box 2, 16.7% chance for box 3. Box 3 gets throw out since all outcomes are silver. 100% results from box one and 50% of results from box two. 1.00*0.5 + 0.5*0.33 = 0.67

How am I hand waiving?

1 box has 3 gold balls the other has 1. If I picked a gold ball, it's a 3 out of 4 chance it came from the box with 3 gold balls. Now, when I pick my second ball, there are only 2 outcomes. It will either be gold (75% of the time) or silver (25% of the time).

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Here's my logic:

I know I'm going to pull a gold so, 50% chance for Box 1, 33% chance for box 2, 16.7% chance for box 3. Box 3 gets throw out since all outcomes are silver. 100% results from box one and 50% of results from box two. 1.00*0.5 + 0.5*0.33 = 0.67

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The math is sound.

Your question and the OP are basically if/then statements. If we've chosen gold, then what are the odds the next ball is gold. Well, first we need to figure out how many ways we can choose gold (3 in the OP, 6 in your example). Then we need to assign the probability of the next ball being gold for each of the possible ways we can start with a gold ball. In the OP, G1 and G2 have a 100% chance EACH of having a gold ball come next. G3 has a 0% chance. Those are 3 possible outcomes, and 67% of them result in a gold ball coming next.

In your example, G1, G2, and G3 have a 100% chance each of having a gold ball come next. G4 and G5 have a 50% chance each, and G6 has a 0% chance. 4 of the 6 possible outcomes result in a gold ball being picked again, ergo, 67% chance of pulling another gold on the next pull.