From a pack of 52 cards, a card is chosen at random. Find the probability that the chosen card is :
(i) A black king.
(ii) Neither a heart nor a king.

Options

A. `{:(""(i),(ii)),(1/26,7/13):}`
B. `{:(""(i),(ii)),(1/13,9/13):}`
C. `{:(""(i),(ii)),(1/13,7/13):}`
D. `{:(""(i),(ii)),(1/26,9/13):}`

Answer

To solve the problem, we need to find the probabilities for two scenarios involving a standard deck of 52 playing cards.

### Step-by-Step Solution:

**(i) Probability of drawing a black king:**

1. **Identify the total number of cards in the deck:**
- A standard deck has 52 cards.

2. **Identify the number of black kings in the deck:**
- There are 2 black kings in a deck: the King of Spades and the King of Clubs.

3. **Calculate the probability:**
- The probability \( P(E1) \) of drawing a black king is given by the formula:
\[ P(E1) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{52} \]

4. **Simplify the fraction:**
- Simplifying \( \frac{2}{52} \) gives:
\[ P(E1) = \frac{1}{26} \]

**(ii) Probability of drawing a card that is neither a heart nor a king:**

1. **Identify the total number of cards in the deck:**
- Again, there are 52 cards in total.

2. **Identify the number of hearts and kings:**
- There are 13 hearts in the deck.
- There are 4 kings in total (1 king in each suit).

3. **Count the overlapping card (the King of Hearts):**
- The King of Hearts is counted in both the hearts and the kings. Therefore, we need to adjust for this overlap.
- Total cards that are either hearts or kings:
\[ 13 \text{ (hearts)} + 4 \text{ (kings)} - 1 \text{ (King of Hearts)} = 16 \]

4. **Calculate the number of cards that are neither hearts nor kings:**
- The number of cards that are neither hearts nor kings is:
\[ 52 - 16 = 36 \]

5. **Calculate the probability:**
- The probability \( P(E2) \) of drawing a card that is neither a heart nor a king is:
\[ P(E2) = \frac{36}{52} \]

6. **Simplify the fraction:**
- Simplifying \( \frac{36}{52} \) gives:
\[ P(E2) = \frac{9}{13} \]

### Final Answers:
- (i) Probability of drawing a black king: \( \frac{1}{26} \)
- (ii) Probability of drawing a card that is neither a heart nor a king: \( \frac{9}{13} \)