Four cards are drawn at random from a pack of 52 playing cards , Find the probability of getting
two red cards and two black cards ,

Answer

To solve the problem of finding the probability of drawing 2 red cards and 2 black cards from a pack of 52 playing cards, we can follow these steps:

### Step 1: Determine the total number of ways to select 4 cards from 52.
The total number of ways to select 4 cards from a pack of 52 cards is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

\[ \text{Total ways to select 4 cards} = \binom{52}{4} \]

### Step 2: Determine the number of red and black cards.
In a standard deck of playing cards, there are 26 red cards and 26 black cards.

### Step 3: Calculate the number of ways to select 2 red cards from 26 red cards.
The number of ways to select 2 red cards from 26 red cards is:

\[ \text{Ways to select 2 red cards} = \binom{26}{2} \]

### Step 4: Calculate the number of ways to select 2 black cards from 26 black cards.
Similarly, the number of ways to select 2 black cards from 26 black cards is:

\[ \text{Ways to select 2 black cards} = \binom{26}{2} \]

### Step 5: Calculate the total number of favorable outcomes.
The total number of favorable outcomes for selecting 2 red cards and 2 black cards is the product of the ways to select the red cards and the ways to select the black cards:

\[ \text{Total favorable outcomes} = \binom{26}{2} \times \binom{26}{2} \]

### Step 6: Calculate the probability.
The probability of drawing 2 red cards and 2 black cards is given by the ratio of the total number of favorable outcomes to the total number of possible outcomes:

\[ \text{Probability} = \frac{\text{Total favorable outcomes}}{\text{Total ways to select 4 cards}} = \frac{\binom{26}{2} \times \binom{26}{2}}{\binom{52}{4}} \]

### Step 7: Calculate the values.
Now we can compute the values of the combinations:

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

2. Calculate \( \binom{26}{2} \):
\[ \binom{26}{2} = \frac{26 \times 25}{2 \times 1} = 325 \]

3. Now substitute these values into the probability formula:
\[ \text{Total favorable outcomes} = 325 \times 325 = 105625 \]
\[ \text{Probability} = \frac{105625}{270725} \]

### Step 8: Simplify the fraction.
To simplify \( \frac{105625}{270725} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD).

After simplification, we find:
\[ \text{Probability} = \frac{325}{833} \]

### Final Answer:
The probability of drawing 2 red cards and 2 black cards from a pack of 52 playing cards is \( \frac{325}{833} \).

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