Two cards are drawn at random from a well-shuffled pack of 52 playing cards. Find the probability that one is a spade and the other is a heart.

Answer

To solve the problem of finding the probability that one card drawn is a spade and the other is a heart from a standard deck of 52 playing cards, we can follow these steps:

### Step 1: Understand the Composition of the Deck
A standard deck of playing cards consists of 52 cards divided into 4 suits: spades, hearts, diamonds, and clubs. Each suit contains 13 cards.

**Hint:** Remember that there are 13 cards in each suit.

### Step 2: Determine the Total Number of Ways to Choose 2 Cards
The total number of ways to choose 2 cards from a deck of 52 cards can be calculated using the combination formula:
\[ \text{Total ways} = \binom{52}{2} = \frac{52 \times 51}{2} = 1326 \]

**Hint:** Use the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\) to find the number of ways to choose cards.

### Step 3: Calculate the Favorable Outcomes
To find the number of favorable outcomes where one card is a spade and the other is a heart, we can consider two scenarios:
1. The first card drawn is a spade and the second card is a heart.
2. The first card drawn is a heart and the second card is a spade.

#### Scenario 1: First Card is a Spade, Second Card is a Heart
- The probability of drawing a spade first:
\[ P(\text{Spade first}) = \frac{13}{52} \]
- After drawing a spade, there are now 51 cards left, and 13 of them are hearts. The probability of drawing a heart next:
\[ P(\text{Heart second | Spade first}) = \frac{13}{51} \]
- Therefore, the combined probability for this scenario is:
\[ P(\text{Spade first, Heart second}) = \frac{13}{52} \times \frac{13}{51} \]

#### Scenario 2: First Card is a Heart, Second Card is a Spade
- The probability of drawing a heart first:
\[ P(\text{Heart first}) = \frac{13}{52} \]
- After drawing a heart, there are still 51 cards left, and 13 of them are spades. The probability of drawing a spade next:
\[ P(\text{Spade second | Heart first}) = \frac{13}{51} \]
- Therefore, the combined probability for this scenario is:
\[ P(\text{Heart first, Spade second}) = \frac{13}{52} \times \frac{13}{51} \]

### Step 4: Combine the Probabilities
Since both scenarios are mutually exclusive, we can add the probabilities:
\[ P(\text{One Spade and One Heart}) = P(\text{Spade first, Heart second}) + P(\text{Heart first, Spade second}) \]
\[ = \left(\frac{13}{52} \times \frac{13}{51}\right) + \left(\frac{13}{52} \times \frac{13}{51}\right) \]
\[ = 2 \times \left(\frac{13}{52} \times \frac{13}{51}\right) \]
\[ = 2 \times \frac{169}{2652} = \frac{338}{2652} \]

### Step 5: Simplify the Probability
Now, simplify \(\frac{338}{2652}\):
\[ \frac{338 \div 2}{2652 \div 2} = \frac{169}{1326} \]
This fraction can be further simplified:
\[ \frac{169}{1326} = \frac{13}{102} \]

### Final Answer
Thus, the probability that one card is a spade and the other is a heart is:
\[ \boxed{\frac{13}{102}} \]