Four cards are successive drawn without replacement from a pack of cards. What is the probability that all the four are aces ?

Answer

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

### Step 1: Identify the total number of cards and aces
In a standard deck of cards, there are a total of 52 cards, which include 4 aces.

### Step 2: Determine the number of favorable outcomes
We want to find the probability of drawing all 4 aces. There is only one way to choose all 4 aces from the 4 available aces, which can be represented mathematically as:
\[ \text{Favorable outcomes} = \binom{4}{4} = 1 \]

### Step 3: Determine the total number of outcomes
Next, we need to determine the total number of ways to draw 4 cards from a deck of 52 cards. This can be calculated using the combination formula:
\[ \text{Total outcomes} = \binom{52}{4} \]
Using the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), we have:
\[ \binom{52}{4} = \frac{52!}{4!(52-4)!} = \frac{52!}{4! \cdot 48!} \]
This simplifies to:
\[ \binom{52}{4} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} \]

### Step 4: Calculate the total outcomes
Calculating the denominator:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
Now calculating the numerator:
\[ 52 \times 51 \times 50 \times 49 = 6497400 \]
Thus, the total outcomes become:
\[ \binom{52}{4} = \frac{6497400}{24} = 270725 \]

### Step 5: Calculate the probability
Now, we can find the probability of drawing all 4 aces:
\[ P(\text{all 4 are aces}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{1}{270725} \]

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