A bag contains one counter, known to be either white or black.

A white counter is put in, the bag shaken, and a counter drawn out,
which proves to be white.

What is now the chance of drawing a white
counter?

(In reply to

re: Without replacement (spoiler) by Brian Smith)

Where to start?

1) No, if a white counter is drawn, the probability that it is the new counter REALLY IS 2/3. But the probability of drawing a white counter is only 3/4, so the probability of drawing the new counter is 3/4 * 2/3 = 1/2, which is what you expect. The probability of pulling the original counter = 3/4 * 1/3 + 1/4 * 1 = 1/2.

2) No, this problem is equivalent to being given a bag with two coins, one is normal heads/tails and the other is double heads. And then being asked what is the probability that heads shows up when a randomly selected coin that has heads on one side is flipped over. There are three heads in the bag, and 2 of them have a head on the flip side, so the probability is 2/3.