colored balls: should you sample with or without replacement?

I found a rather tricky exercise:

Two urns contain red and black balls, all alike except for color. Urn A has 2 reds and 1 black, and Urn B has 101 reds and 100 blacks. An urn is chosen at random, and you win a prize if you correctly name the urn on the basis of the evidence of two balls drawn from it. After the first ball is drawn and its color seen, you can decide whether or not the ball shall be replaced before the second drawing. How do you order the second drawing, and how do you decide on the urn?

How do you approach such a problem?

im pretty sure you would want not to replace a black ball, since if you dont replace it and get a second black, you know that it is urn B with certainty.

for a rigorous, but long, approach you can just evaluate your success probability with every possible strategy, since there is a relatively small number of possible outcomes. Dont forget that whether to replace or not may depend on whether the first ball drawn is red or black.

Only one way to be SURE which urn is which: draw 2 blacks in a row, no replacement;
then you know you're picking from urn#2.
getting urn#2: 1/2 chance
1st pick = black : 100/201
2ns pick = black : 99/200

Quote:

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Originally Posted by SpringFan25

Dont forget that whether to replace or not may depend on whether the first ball drawn is red or black.

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Yes, this is exactly what i was looking for! Thank you