Two cards are drawn from a well-shuffled deck of 52 cards. Find the probability that either both are red or both are kings .

Answer

To solve the problem of finding the probability that either both cards drawn are red or both are kings, we can follow these steps:

### Step 1: Define the Events
Let:
- Event A = Both cards drawn are red.
- Event B = Both cards drawn are kings.

### Step 2: Find the Total Number of Ways to Draw 2 Cards
The total number of ways to draw 2 cards from a deck of 52 cards is given by the combination formula:
\[ \text{Total ways} = \binom{52}{2} = \frac{52 \times 51}{2} = 1326 \]

### Step 3: Calculate the Probability of Event A (Both Cards are Red)
There are 26 red cards in a deck. The number of ways to choose 2 red cards is:
\[ \text{Ways to choose 2 red cards} = \binom{26}{2} = \frac{26 \times 25}{2} = 325 \]
Thus, the probability of event A is:
\[ P(A) = \frac{\text{Ways to choose 2 red cards}}{\text{Total ways}} = \frac{325}{1326} \]

### Step 4: Calculate the Probability of Event B (Both Cards are Kings)
There are 4 kings in a deck. The number of ways to choose 2 kings is:
\[ \text{Ways to choose 2 kings} = \binom{4}{2} = \frac{4 \times 3}{2} = 6 \]
Thus, the probability of event B is:
\[ P(B) = \frac{\text{Ways to choose 2 kings}}{\text{Total ways}} = \frac{6}{1326} \]

### Step 5: Calculate the Probability of the Intersection of Events A and B
The intersection of events A and B (both cards are red kings) can occur only if both cards drawn are the two red kings. There are 2 red kings in the deck. The number of ways to choose 2 red kings is:
\[ \text{Ways to choose 2 red kings} = \binom{2}{2} = 1 \]
Thus, the probability of the intersection is:
\[ P(A \cap B) = \frac{\text{Ways to choose 2 red kings}}{\text{Total ways}} = \frac{1}{1326} \]

### Step 6: Calculate the Probability of A Union B
Using the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the values we calculated:
\[ P(A \cup B) = \frac{325}{1326} + \frac{6}{1326} - \frac{1}{1326} = \frac{325 + 6 - 1}{1326} = \frac{330}{1326} \]

### Step 7: Simplify the Probability
Now, we can simplify \(\frac{330}{1326}\):
\[ P(A \cup B) = \frac{330 \div 6}{1326 \div 6} = \frac{55}{221} \]

### Final Answer
Thus, the probability that either both cards drawn are red or both are kings is:
\[ \boxed{\frac{55}{221}} \]