Four cards are successively drawn without replacement from a deck of 52 playing cards. What is the probability that all the four cards are king?

Answer

To find the probability that all four cards drawn from a deck of 52 playing cards are kings, we can follow these steps:

### Step 1: Determine the probability of drawing the first king.
- There are 4 kings in a deck of 52 cards.
- The probability of drawing the first king (K1) is:
\[ P(K1) = \frac{4}{52} \]

### Step 2: Determine the probability of drawing the second king.
- After drawing the first king, there are now 3 kings left and only 51 cards remaining in the deck.
- The probability of drawing the second king (K2) is:
\[ P(K2) = \frac{3}{51} \]

### Step 3: Determine the probability of drawing the third king.
- After drawing the second king, there are now 2 kings left and only 50 cards remaining in the deck.
- The probability of drawing the third king (K3) is:
\[ P(K3) = \frac{2}{50} \]

### Step 4: Determine the probability of drawing the fourth king.
- After drawing the third king, there is now 1 king left and only 49 cards remaining in the deck.
- The probability of drawing the fourth king (K4) is:
\[ P(K4) = \frac{1}{49} \]

### Step 5: Calculate the total probability of drawing four kings in succession.
- Since these events are dependent, we multiply the probabilities together:
\[ P(\text{All four kings}) = P(K1) \times P(K2) \times P(K3) \times P(K4) \]
\[ P(\text{All four kings}) = \frac{4}{52} \times \frac{3}{51} \times \frac{2}{50} \times \frac{1}{49} \]

### Step 6: Simplify the expression.
- Now, we can simplify the expression:
\[ P(\text{All four kings}) = \frac{4 \times 3 \times 2 \times 1}{52 \times 51 \times 50 \times 49} \]
\[ = \frac{24}{270725} \]
\[ = \frac{1}{11230.2083} \approx \frac{1}{270725} \]

### Final Answer:
The probability that all four cards drawn are kings is:
\[ \frac{1}{270725} \]