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 any one suit, e.g., a diamond, heart, spade, or club?

If two cards are randomly selected, what is the probability of drawing a face card first, followed by drawing a numbered card?

If two cards are randomly selected, what is the probability of drawing any one suit first, followed by drawing a face card?

Let's go through each of the probability questions step by step.

### 1. Probability of Drawing Any One Suit
A standard deck has 52 cards, divided equally among the four suits (diamonds, hearts, spades, and clubs), so each suit has 13 cards.

The probability of drawing any one suit, such as a diamond, is the ratio of the number of cards in that suit to the total number of cards in the deck:
\[
P(\text{any one suit}) = \frac{\text{Number of cards in one suit}}{\text{Total number of cards}} = \frac{13}{52}
\]
Now simplify the fraction:
\[
P(\text{any one suit}) = \frac{1}{4}
\]

### 2. Probability of Drawing a Face Card First, Followed by a Numbered Card
In a deck, there are 12 face cards (3 face cards per suit: Jack, Queen, and King). There are 40 numbered cards (4 suits × 10 numbered cards: 2 through 10).

If two cards are drawn without replacement, the probabilities are calculated step by step.

- **Step 1: Probability of drawing a face card first.** There are 12 face cards out of 52 cards.
  \[
  P(\text{Face card first}) = \frac{12}{52} = \frac{3}{13}
  \]

- **Step 2: Probability of drawing a numbered card second.** After drawing a face card, 51 cards remain in the deck, and 40 of those are numbered cards.
  \[
  P(\text{Numbered card second}) = \frac{40}{51}
  \]

- **Overall probability:** Multiply the two probabilities together:
  \[
  P(\text{Face card first, numbered card second}) = \frac{3}{13} \times \frac{40}{51} = \frac{120}{663}
  \]

Thus, the probability is:
\[
P(\text{Face card first, numbered card second}) = \frac{120}{663}
\]

### 3. Probability of Drawing Any One Suit First, Followed by a Face Card
Again, we proceed step by step for this two-card draw.

- **Step 1: Probability of drawing any one suit first.** There are 13 cards of any one suit (e.g., diamonds) out of 52.
  \[
  P(\text{One suit first}) = \frac{13}{52} = \frac{1}{4}
  \]

- **Step 2: Probability of drawing a face card second.** After drawing one card, 51 cards remain, and 12 of those are face cards.
  \[
  P(\text{Face card second}) = \frac{12}{51} = \frac{4}{17}
  \]

- **Overall probability:** Multiply the two probabilities together:
  \[
  P(\text{One suit first, face card second}) = \frac{1}{4} \times \frac{4}{17} = \frac{4}{68} = \frac{1}{17}
  \]

Thus, the probability is:
\[
P(\text{One suit first, face card second}) = \frac{1}{17}
\]

### Summary of Results:
1. Probability of drawing any one suit: \(\frac{1}{4}\)
2. Probability of drawing a face card first and a numbered card second: \(\frac{120}{663}\)
3. Probability of drawing any one suit first and a face card second: \(\frac{1}{17}\)

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Would you like more details on any of these? 

Here are 5 related questions you can explore:
1. What is the probability of drawing two face cards in a row?
2. If two cards are drawn, what is the probability of getting two cards of the same suit?
3. What is the probability of drawing a numbered card first, followed by a face card?
4. How does the probability change if you are allowed to replace the card after drawing it?
5. What is the probability of drawing two diamonds in a row?

**Tip:** When calculating probabilities of multiple events, remember to multiply the individual probabilities for each event, considering whether or not you are replacing the drawn cards.

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 any one suit, e.g., a diamond, heart, spade, or club? If two cards are randomly selected, what is the probability of drawing a face card first, followed by drawing a numbered card? If two cards are randomly selected, what is the probability of drawing any one suit first, followed by drawing a face card?

Explore the probability of drawing specific cards from a 52-card deck. Learn how to calculate the probability of drawing one suit, a face card followed by a numbered card, or a suit followed by a face card. Detailed solutions with step-by-step explanations, suitable for Grades 9-11.

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