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 can break it down step by step.

### Part (I): First Card is Replaced Before the Second is Taken

1. **Identify Total Outcomes**:
- A standard deck has 52 cards.

2. **Identify Favorable Outcomes for the First Draw**:
- There are 13 spades in the deck.
- The probability of drawing a spade on the first draw is:
\[ P(\text{First Spade}) = \frac{13}{52} = \frac{1}{4} \]

3. **Replacement of the First Card**:
- Since the first card is replaced, the total number of cards remains 52 for the second draw.

4. **Identify Favorable Outcomes for the Second Draw**:
- The probability of drawing a spade on the second draw remains the same:
\[ P(\text{Second Spade}) = \frac{13}{52} = \frac{1}{4} \]

5. **Calculate Total Probability**:
- Since the draws are independent (due to replacement), the total probability of both events happening is the product of their individual probabilities:
\[ P(\text{Two Spades with Replacement}) = P(\text{First Spade}) \times P(\text{Second Spade}) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \]

### Part (II): First Card is Not Replaced

1. **Identify Total Outcomes**:
- Again, we start with 52 cards.

2. **Identify Favorable Outcomes for the First Draw**:
- The probability of drawing a spade on the first draw remains:
\[ P(\text{First Spade}) = \frac{13}{52} = \frac{1}{4} \]

3. **No Replacement of the First Card**:
- After drawing the first spade, we do not replace it. Thus, there are now 12 spades left in a total of 51 cards.

4. **Identify Favorable Outcomes for the Second Draw**:
- The probability of drawing a spade on the second draw is now:
\[ P(\text{Second Spade}) = \frac{12}{51} \]

5. **Calculate Total Probability**:
- The total probability of both events happening is:
\[ P(\text{Two Spades without Replacement}) = P(\text{First Spade}) \times P(\text{Second Spade}) = \frac{1}{4} \times \frac{12}{51} = \frac{12}{204} = \frac{1}{17} \]

### Final Answers:
- (I) Probability of drawing two spades with replacement: \(\frac{1}{16}\)
- (II) Probability of drawing two spades without replacement: \(\frac{1}{17}\)