Find the probability of drawing two spades from a well shuffled pack of 52 cards if :
(I) the first card is replaced before the second is taken (II) First card is not replaced

Answer

To solve the problem of finding the probability of drawing two spades from a well-shuffled pack of 52 cards under two different scenarios, we will break it down step by step.

### Step 1: Understand the total number of cards and spades
A standard deck of cards contains 52 cards, out of which there are 13 spades.

### Step 2: Case I - First card is replaced before the second is taken
1. **Probability of drawing the first spade**:
- The number of favorable outcomes (drawing a spade) = 13
- Total outcomes = 52
- Probability of drawing the first spade = \( P(A) = \frac{13}{52} = \frac{1}{4} \)

2. **Probability of drawing the second spade (after replacement)**:
- Since the first card is replaced, the total number of cards remains 52 and the number of spades remains 13.
- Probability of drawing the second spade = \( P(B) = \frac{13}{52} = \frac{1}{4} \)

3. **Combined probability for both events**:
- Since the two events are independent (due to replacement), the combined probability is:
\[ P(A \text{ and } B) = P(A) \times P(B) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \]

### Step 3: Case II - First card is not replaced
1. **Probability of drawing the first spade**:
- As in the first case, the probability of drawing the first spade remains the same:
\[ P(A) = \frac{13}{52} = \frac{1}{4} \]

2. **Probability of drawing the second spade (without replacement)**:
- After drawing the first spade, there are now 12 spades left and only 51 cards remaining.
- Probability of drawing the second spade = \( P(B|A) = \frac{12}{51} \)

3. **Combined probability for both events**:
- The combined probability is:
\[ P(A \text{ and } B) = P(A) \times P(B|A) = \frac{1}{4} \times \frac{12}{51} = \frac{12}{204} = \frac{1}{17} \]

### Final Answers:
- **Case I (with replacement)**: The probability of drawing two spades is \( \frac{1}{16} \).
- **Case II (without replacement)**: The probability of drawing two spades is \( \frac{1}{17} \).

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