This is a classic problem in probability that can be solved using conditional probability.

Let's summarize the situation:

• There are three cards:
  • Card 1: Red on both sides (denote as RR)
  • Card 2: Black on both sides (denote as BB)
  • Card 3: Red on one side, Black on the other side (denote as RB)

We are asked to find the probability that, given that a red side is facing up, the other side of the card is also red. This means we are looking for the probability that the chosen card is the RR card, given that we observe a red side facing up.

Step-by-Step Solution:

1. Total possible outcomes (red facing up):
   Each card has two sides. We need to count how many "red" sides can possibly face up:

   • Card 1 (RR): Both sides are red, so there are 2 red sides.
   • Card 2 (BB): There are no red sides.
   • Card 3 (RB): There is 1 red side.

   So, in total, there are $2 + 0 + 1 = 3$ possible ways for a red side to face up.

2. Favorable outcomes (red on the other side):
   We want to know how many of these cases involve the other side also being red. The only card for which both sides are red is Card 1 (RR). Since both sides are red, there are 2 favorable outcomes for this card.

3. Calculate the probability:
   The probability we want is the number of favorable outcomes divided by the total possible outcomes: $P(\text{RR} \mid \text{red up}) = \frac{\text{favorable outcomes}}{\text{total red outcomes}} = \frac{2}{3}$

Thus, the probability that the other side is also red, given that the side facing up is red, is $\frac{2}{3}$.

Do you want more details or have any questions?

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Related Questions:

1. What is conditional probability, and how is it calculated in general?
2. If the side facing up is black, what is the probability that the other side is red?
3. How does Bayes' Theorem relate to solving this type of probability problem?
4. What is the probability that the card chosen is the one with red and black sides, given that red is facing up?
5. Can this problem be solved using a probability tree? How?

Tip: When dealing with conditional probability problems, always clarify the outcomes you're given and the specific outcomes you're asked to find.