Question

A card is drawn from a deck of 52 cards. What is the probability that it is a picture card​ (Jack, Queen,​ King) or a diamond​?
Question content area bottom
Part 1

Answer 1

To find the probability that a card drawn from a standard deck of 52 cards is either a picture card (Jack, Queen, King) or a diamond, we can use the principle of inclusion-exclusion.

### Step 1: Determine the total number of favorable outcomes.

**1. Count the picture cards:**
- There are 3 picture cards in each suit: Jack, Queen, and King.
- Since there are 4 suits, the total number of picture cards is:
\[
3 \text{ cards/suit} \times 4 \text{ suits} = 12 \text{ picture cards}
\]

**2. Count the diamonds:**
- There are 13 diamonds in total (one for each rank).

**3. Overlap:**
- The diamonds include 3 picture cards (Jack of diamonds, Queen of diamonds, King of diamonds).

### Step 2: Use inclusion-exclusion to find the total number of favorable outcomes.

Let \( A \) be the event of drawing a picture card and \( B \) be the event of drawing a diamond.

Using the inclusion-exclusion principle:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

- \( P(A) \) (number of picture cards) = 12
- \( P(B) \) (number of diamonds) = 13
- \( P(A \cap B) \) (number of picture cards that are also diamonds) = 3

### Total favorable outcomes:
\[
P(A \cup B) = 12 + 13 - 3 = 22
\]

### Step 3: Find the probability.

The total number of cards in the deck is 52. Thus, the probability is:
\[
P(A \cup B) = \frac{\text{Total favorable outcomes}}{\text{Total outcomes}} = \frac{22}{52}
\]

### Step 4: Simplify the probability.

To simplify \( \frac{22}{52} \):
\[
\frac{22 \div 2}{52 \div 2} = \frac{11}{26}
\]

### Final Answer:
The probability that the card drawn is a picture card or a diamond is \(\frac{11}{26}\).