Probability of Drawing 2 Aces and 2 Kings

Problem

What is the probability of drawing exactly $2$ aces and $2$ kings from a standard $52$-card deck?

Step 1: Count the Good Hands

We want hands that have exactly $2$ aces and $2$ kings.

There are $4$ aces in the deck, and we choose $2$ of them:

$$\binom{4}{2} = 6$$

There are also $4$ kings in the deck, and we choose $2$ of them:

$$\binom{4}{2} = 6$$

So the number of good hands is:

$$6 \times 6 = 36$$

Step 2: Count All Possible Hands

Now we count all the ways to draw any $4$ cards from a $52$-card deck:

$$\binom{52}{4} = 270{,}725$$

Step 3: Find the Probability

The probability is:

$$\frac{\text{good hands}}{\text{all hands}} = \frac{36}{270{,}725}$$

So:

$$\frac{36}{270{,}725} \approx 0.000133$$

Step 4: Final Answer

The probability of drawing exactly $2$ aces and $2$ kings is:

$$\boxed{\frac{36}{270{,}725} \approx 0.000133}$$

That means it is very unlikely. Out of many, many $4$-card hands, only a tiny number will have exactly $2$ aces and $2$ kings.