Statistics question Conditional Probability Question: Of three cards, one is painted red on both sides; one is painted black on both sides; and one is painted red on one side and black on the other. A card is randomly chosen and placed on a table. If the side facing up is red, what is the probability that the other side is also red? My Attempt: number designates side and letter the colour Card 1: R1 R2 Card 2: B1 B2 Card 3: R1 B2 P(R2|R1) = P(R1R2)/P(R1) P(R2|R1) = (1/3)/(1/2) P(R2|R1) = 2/3 I did this in a conditional probability manner but my instinct says the answer should just be 1/2...

Your intuition is that each card has an equal probability of being chosen, and this is true. Yet you must consider that the cards have an not so equal probability of being chosen _and_ showing their red side.

Instead, observe that if I select a card, then select a side to show, both choices without bias, then _every side_ has an equal probability of being the one shown.

Now, when given that the side shown is red, the three _red sides_ still have equal probability of being that one shown. However only of them have a red otherside. The third red side has a green otherside.

Therefore there must be a conditional probability of $2/3$ for the otherside of the side shown to be red _when given_ that the side shown is itself red.