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

Answer

To find the probability of drawing four cards from a pack of 52 playing cards such that all four cards are of the same suit, we can follow these steps:

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

\[ \text{Total outcomes} = \binom{52}{4} = \frac{52!}{4!(52-4)!} = \frac{52!}{4! \cdot 48!} \]

### Step 2: Determine the number of favorable outcomes.
To find the number of ways to draw 4 cards of the same suit, we first note that there are 4 suits in a deck (hearts, diamonds, clubs, spades), and each suit has 13 cards.

For each suit, the number of ways to choose 4 cards from 13 is:

\[ \text{Favorable outcomes for one suit} = \binom{13}{4} = \frac{13!}{4!(13-4)!} = \frac{13!}{4! \cdot 9!} \]

Since there are 4 suits, the total number of favorable outcomes is:

\[ \text{Total favorable outcomes} = 4 \times \binom{13}{4} \]

### Step 3: Calculate the probability.
The probability \( P \) of drawing 4 cards of the same suit is given by the ratio of the number of favorable outcomes to the total number of outcomes:

\[ P = \frac{\text{Total favorable outcomes}}{\text{Total outcomes}} = \frac{4 \times \binom{13}{4}}{\binom{52}{4}} \]

### Step 4: Compute the values.
Now we compute the values of the combinations:

1. Calculate \( \binom{13}{4} \):

\[ \binom{13}{4} = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} = \frac{17160}{24} = 715 \]

2. Calculate \( \binom{52}{4} \):

\[ \binom{52}{4} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = \frac{67605252}{24} = 282475249 \]

3. Substitute these values into the probability formula:

\[ P = \frac{4 \times 715}{\binom{52}{4}} = \frac{2860}{270725} \]

### Step 5: Simplify the probability.
Now we simplify the fraction:

\[ P = \frac{2860}{270725} \approx 0.01057 \]

Thus, the probability of drawing 4 cards of the same suit is approximately \( 0.01057 \) or \( \frac{286}{270725} \).