Four cards are drawn at random from a pack of 52 playing cards , Find the probability of getting
all cards of the same colour ,

Answer

To solve the problem of finding the probability of drawing four cards of the same color from a standard deck of 52 playing cards, we can follow these steps:

### Step 1: Determine the total number of ways to select 4 cards from 52 cards.
The total number of ways to select 4 cards from a deck of 52 cards is given by the combination formula:

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

### Step 2: Determine the number of red and black cards in the deck.
In a standard deck of cards, there are:
- 26 red cards (13 hearts and 13 diamonds)
- 26 black cards (13 clubs and 13 spades)

### Step 3: Calculate the number of ways to select 4 red cards.
The number of ways to select 4 cards from the 26 red cards is:

\[ \text{Ways to select 4 red cards} = \binom{26}{4} \]

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

\[ \text{Ways to select 4 black cards} = \binom{26}{4} \]

### Step 5: Calculate the total number of favorable outcomes.
The total number of favorable outcomes for getting 4 cards of the same color (either all red or all black) is:

\[ \text{Favorable outcomes} = \binom{26}{4} + \binom{26}{4} = 2 \cdot \binom{26}{4} \]

### Step 6: Calculate the probability.
The probability of drawing 4 cards of the same color is given by the ratio of the number of favorable outcomes to the total number of outcomes:

\[ P(\text{same color}) = \frac{\text{Favorable outcomes}}{\text{Total ways}} = \frac{2 \cdot \binom{26}{4}}{\binom{52}{4}} \]

### Step 7: Substitute the values and simplify.
Now we can calculate the combinations:

\[ \binom{26}{4} = \frac{26 \times 25 \times 24 \times 23}{4 \times 3 \times 2 \times 1} = 14950 \]
\[ \binom{52}{4} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270725 \]

Now substituting these values into the probability formula:

\[ P(\text{same color}) = \frac{2 \cdot 14950}{270725} = \frac{29900}{270725} \]

### Step 8: Simplify the fraction.
After simplifying, we find:

\[ P(\text{same color}) \approx \frac{92}{833} \]

Thus, the final answer is:

\[ \text{Probability of getting all cards of the same color} = \frac{92}{833} \]

---