What is the conditional probability of there being at least 3 kings in a random hand of 13 cards, picked out of a deck of 52 cards, given that there are at least 2 kings in the hand?

To solve this problem, we'll apply the concept of conditional probability.

### Given:
- A standard deck of 52 cards.
- A hand of 13 cards.
- We need to find the conditional probability that there are at least 3 kings in the hand, given that there are at least 2 kings in the hand.

### Conditional Probability Formula:
The conditional probability \( P(A \mid B) \) is given by:
\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]
Where:
- \( A \) is the event "there are at least 3 kings in the hand."
- \( B \) is the event "there are at least 2 kings in the hand."

### Step 1: Calculate \( P(A \cap B) \)
The event \( A \cap B \) corresponds to the situation where there are at least 3 kings in the hand.

Let’s break it down:
- The number of ways to choose 3 kings out of 4: \( \binom{4}{3} = 4 \).
- The number of ways to choose the remaining 10 cards from the 48 non-kings: \( \binom{48}{10} \).
- The number of ways to choose 4 kings out of 4: \( \binom{4}{4} = 1 \).
- The number of ways to choose the remaining 9 cards from the 48 non-kings: \( \binom{48}{9} \).

Thus, the number of favorable outcomes for \( A \cap B \) is:
\[
\text{Number of favorable outcomes} = \binom{4}{3} \times \binom{48}{10} + \binom{4}{4} \times \binom{48}{9}
\]

### Step 2: Calculate \( P(B) \)
The event \( B \) corresponds to the situation where there are at least 2 kings in the hand.

Let’s break it down:
- The number of ways to choose 2 kings out of 4: \( \binom{4}{2} = 6 \).
- The number of ways to choose the remaining 11 cards from the 48 non-kings: \( \binom{48}{11} \).

Thus, the number of favorable outcomes for \( B \) is:
\[
\text{Number of favorable outcomes} = \binom{4}{2} \times \binom{48}{11} + \text{(Outcomes for \( A \cap B \))}
\]
Since the event \( A \cap B \) is a subset of \( B \), you add these outcomes to \( P(B) \).

### Step 3: Simplify and Find \( P(A \mid B) \)

Given the calculations for \( P(A \cap B) \) and \( P(B) \), the conditional probability \( P(A \mid B) \) is the ratio of these two.

Let me calculate these values.The conditional probability that there are at least 3 kings in a hand of 13 cards, given that there are at least 2 kings in the hand, is approximately **0.1704**.

Would you like more details, or do you have any questions?

Here are 5 related questions for further exploration:
1. What is the probability of having exactly 3 kings in a hand of 13 cards?
2. How would the probability change if the hand size was 10 cards instead of 13?
3. What is the expected number of kings in a random hand of 13 cards?
4. How does the probability of getting at least one king in a hand of 13 cards compare to the result above?
5. How would the approach change if we were using a deck with jokers included?

**Tip:** Conditional probability is a powerful tool in situations where additional information is provided, allowing for a more focused probability calculation.

Calculate the conditional probability of having at least 3 kings in a random hand of 13 cards, given that there are at least 2 kings in the hand. Learn the step-by-step process using combinatorics and probability theory. Suitable for advanced high school students and beyond.

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