Four cards are drawn from a full pack of cards . Find the probability that
there are two spades and two hearts,

Answer

To find the probability of drawing 4 cards from a full pack of 52 cards such that there are exactly 2 spades and 2 hearts, we can follow these steps:

### Step 1: Calculate the Total Possible Outcomes
The total number of ways to draw 4 cards from 52 cards is given by the combination formula \( \binom{n}{r} \), which represents the number of ways to choose \( r \) items from \( n \) items without regard to the order of selection.

\[ \text{Total outcomes} = \binom{52}{4} \]

### Step 2: Calculate the Number of Favorable Outcomes
We need to find the number of ways to choose 2 spades and 2 hearts.

1. **Choosing 2 Spades**: There are 13 spades in a deck, so the number of ways to choose 2 spades from 13 is:
\[ \text{Ways to choose 2 spades} = \binom{13}{2} \]

2. **Choosing 2 Hearts**: Similarly, there are also 13 hearts, so the number of ways to choose 2 hearts from 13 is:
\[ \text{Ways to choose 2 hearts} = \binom{13}{2} \]

3. **Total Favorable Outcomes**: Since we need both events (choosing 2 spades and 2 hearts), we multiply the number of ways to choose spades and hearts:
\[ \text{Favorable outcomes} = \binom{13}{2} \times \binom{13}{2} \]

### Step 3: Calculate the Probability
The probability \( P \) of drawing 2 spades and 2 hearts is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{\binom{13}{2} \times \binom{13}{2}}{\binom{52}{4}} \]

### Step 4: Calculate the Values
1. Calculate \( \binom{13}{2} \):
\[ \binom{13}{2} = \frac{13 \times 12}{2 \times 1} = 78 \]

2. Calculate \( \binom{52}{4} \):
\[ \binom{52}{4} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270725 \]

3. Substitute the values into the probability formula:
\[ P = \frac{78 \times 78}{270725} = \frac{6084}{270725} \]

### Step 5: Simplify the Probability
Now we can simplify \( \frac{6084}{270725} \) if possible.

After simplification, we find:
\[ P \approx 0.0225 \]

Thus, the probability of drawing 2 spades and 2 hearts from a pack of 52 cards is approximately \( 0.0225 \).