Using a standard 52-card deck, find the following probabilities. Leave your answer as a reduced fraction.
If a card is randomly selected, what is the probability that the card is a diamond?

If two cards are randomly selected, what is the probability of drawing a king first, followed by drawing a 2?

If two cards are randomly selected, what is the probability of drawing a heart first, followed by drawing a queen?

Let's tackle each probability problem step by step.

### 1. **Probability that the card is a diamond**:
In a standard 52-card deck, there are 13 diamonds (one for each rank: Ace, 2, 3, ..., King). Therefore, the probability of randomly selecting a diamond is:

\[
P(\text{diamond}) = \frac{\text{number of diamonds}}{\text{total number of cards}} = \frac{13}{52} = \frac{1}{4}
\]

### 2. **Probability of drawing a king first, followed by drawing a 2**:
- The number of kings in a deck is 4.
- After drawing a king, there are 51 cards left, and the number of 2's in the deck is 4.

The probability of drawing a king first and then drawing a 2 is the product of these two independent probabilities:

\[
P(\text{king, then 2}) = \frac{\text{number of kings}}{\text{total cards}} \times \frac{\text{number of 2's}}{\text{remaining cards}} = \frac{4}{52} \times \frac{4}{51} = \frac{16}{2652} = \frac{4}{663}
\]

### 3. **Probability of drawing a heart first, followed by drawing a queen**:
- The number of hearts in the deck is 13.
- After drawing a heart, 51 cards remain, and the number of queens is 4.

The probability of drawing a heart first and then drawing a queen is:

\[
P(\text{heart, then queen}) = \frac{\text{number of hearts}}{\text{total cards}} \times \frac{\text{number of queens}}{\text{remaining cards}} = \frac{13}{52} \times \frac{4}{51} = \frac{52}{2652} = \frac{1}{51}
\]

### Summary of answers:
1. Probability of drawing a diamond: \( \frac{1}{4} \)
2. Probability of drawing a king first, followed by drawing a 2: \( \frac{4}{663} \)
3. Probability of drawing a heart first, followed by drawing a queen: \( \frac{1}{51} \)

Would you like more details or have any questions about these calculations?

Here are 5 related questions to expand your understanding:
1. What is the probability of drawing a spade from a standard deck?
2. What is the probability of drawing two aces in a row?
3. How does the probability change if the first card is not replaced before the second draw?
4. What is the probability of drawing a face card (Jack, Queen, King) from the deck?
5. What is the probability of drawing a red card first and then a black card?

**Tip**: When calculating probabilities involving sequential events without replacement, always remember to adjust the total number of remaining cards for each subsequent draw.

Using a standard 52-card deck, find the following probabilities. Leave your answer as a reduced fraction. If a card is randomly selected, what is the probability that the card is a diamond? If two cards are randomly selected, what is the probability of drawing a king first, followed by drawing a 2? If two cards are randomly selected, what is the probability of drawing a heart first, followed by drawing a queen?

Learn how to calculate probabilities using a standard 52-card deck. This guide covers finding the probability of drawing a diamond, drawing a king followed by a 2, and drawing a heart followed by a queen. Perfect for students in Grades 9-12 studying probability and combinatorics.

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