Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Then the mean of the number of kings is :

Question

Accepted Answer

To determine the mean of the number of kings when two cards are drawn simultaneously from a well-shuffled deck of 52 cards, we need to calculate the expected value of the number of kings drawn.

### Explanation:

1. **Total Number of Cards and Kings:**
   - A standard deck has 52 cards, including 4 kings.

2. **Possible Outcomes:**
   - When two cards are drawn, there are several possible outcomes regarding the number of kings:
     - 0 kings
     - 1 king
     - 2 kings

3. **Calculating Probabilities:**
   - **Probability of drawing 0 kings:**
     - The number of ways to draw 2 non-king cards from the 48 non-king cards is $\binom{48}{2}$.
     - Total ways to draw any 2 cards from 52 is $\binom{52}{2}$.
     - Probability of 0 kings = $\frac{\binom{48}{2}}{\binom{52}{2}}$.

- **Probability of drawing 1 king:**
     - Choose 1 king from 4 kings: $\binom{4}{1}$.
     - Choose 1 non-king from 48 non-kings: $\binom{48}{1}$.
     - Probability of 1 king = $\frac{\binom{4}{1} \times \binom{48}{1}}{\binom{52}{2}}$.

- **Probability of drawing 2 kings:**
     - Choose 2 kings from 4 kings: $\binom{4}{2}$.
     - Probability of 2 kings = $\frac{\binom{4}{2}}{\binom{52}{2}}$.

4. **Expected Value (Mean) Calculation:**
   - The expected value $E(X)$ is calculated as:
     $$
     E(X) = 0 \times P(0 \text{ kings}) + 1 \times P(1 \text{ king}) + 2 \times P(2 \text{ kings})
     $$
   - Substituting the probabilities:
     $$
     E(X) = 0 \times \frac{\binom{48}{2}}{\binom{52}{2}} + 1 \times \frac{\binom{4}{1} \times \binom{48}{1}}{\binom{52}{2}} + 2 \times \frac{\binom{4}{2}}{\binom{52}{2}}
     $$

5. **Simplifying the Expression:**
   - Calculate each component:
     - $\binom{48}{2} = 1128$
     - $\binom{52}{2} = 1326$
     - $\binom{4}{1} = 4$
     - $\binom{48}{1} = 48$
     - $\binom{4}{2} = 6$

- Substituting these values:
     $$
     E(X) = 0 + \frac{4 \times 48}{1326} + \frac{2 \times 6}{1326}
     $$
     $$
     E(X) = \frac{192}{1326} + \frac{12}{1326}
     $$
     $$
     E(X) = \frac{204}{1326}
     $$
     $$
     E(X) = \frac{34}{221}
     $$

6. **Conclusion:**
   - The mean number of kings drawn is $\frac{34}{221}$, which corresponds to Option 4.

### Why Other Options are Incorrect:

- **Option 1: $\frac{1}{221}$**
  - This value is significantly lower than the calculated mean and does not match the expected value derived from the probabilities.

- **Option 2: $\frac{2}{221}$**
  - This value is also lower than the calculated mean and does not align with the expected value.

- **Option 3: $\frac{32}{221}$**
  - Although closer, this value is still not equal to the calculated mean of $\frac{34}{221}$.

Thus, the correct answer is indeed Option 4: $\frac{34}{221}$.