you have a well-shuffled standard deck of cards from which you draw without replacement two cards. What is the probability that both cards are red given that at least one of the cards is red?

To solve this problem, we’ll use conditional probability.

Let:
- \( A \) = the event that both cards are red.
- \( B \) = the event that at least one of the cards is red.

We are asked to find \( P(A|B) \), which is the probability that both cards are red given that at least one of the cards is red.

Using the definition of conditional probability, we have:
\[
P(A|B) = \frac{P(A \cap B)}{P(B)}
\]

### Step 1: Calculate \( P(A \cap B) \)
\( P(A \cap B) \) is simply \( P(A) \) because if both cards are red, then the condition of having at least one red card is automatically satisfied.

- There are 26 red cards in a 52-card deck.
- The probability that the first card drawn is red is \( \frac{26}{52} = \frac{1}{2} \).
- If the first card is red, there are now 25 red cards out of the remaining 51 cards. The probability that the second card is red is \( \frac{25}{51} \).

So:
\[
P(A) = \frac{1}{2} \times \frac{25}{51} = \frac{25}{102}
\]

Thus, \( P(A \cap B) = \frac{25}{102} \).

### Step 2: Calculate \( P(B) \)
Next, we calculate \( P(B) \), the probability that at least one of the two drawn cards is red.

To find \( P(B) \), it’s easier to first calculate the probability of the complement event (i.e., the probability that neither card is red) and then subtract this from 1.

- The probability that the first card is black is \( \frac{26}{52} = \frac{1}{2} \).
- Given that the first card is black, the probability that the second card is also black is \( \frac{25}{51} \).

So, the probability that both cards are black (i.e., neither card is red) is:
\[
P(\text{both black}) = \frac{1}{2} \times \frac{25}{51} = \frac{25}{102}
\]

Thus:
\[
P(B) = 1 - \frac{25}{102} = \frac{77}{102}
\]

### Step 3: Calculate \( P(A|B) \)
Now we substitute into the formula for conditional probability:
\[
P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{25}{102}}{\frac{77}{102}} = \frac{25}{77}
\]

### Final Answer
The probability that both cards are red given that at least one of the cards is red is:
\[
\boxed{\frac{25}{77}}
\]

Would you like a detailed breakdown of each step or have any questions?

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**Related Questions for Further Exploration:**
1. What is the probability of drawing two black cards given that at least one card is black?
2. How would the probability change if we drew three cards instead of two?
3. What is the probability that exactly one of the two drawn cards is red?
4. If you draw three cards, what is the probability that all of them are red?
5. How does the probability change if we know the first card is red?

**Tip:** For conditional probability problems, it’s often easier to start by calculating the complement of the event.

You have a well-shuffled standard deck of cards from which you draw without replacement two cards. What is the probability that both cards are red given that at least one of the cards is red?

Explore the probability of drawing two red cards given that at least one is red from a well-shuffled standard deck. This solution covers concepts of conditional probability, including step-by-step calculations of P(A|B) and the use of complementary events. Ideal for high school students in Grades 10-12.

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