A card is drawn at random from a pack of `52` cards. Find the probability that the card drawn is:
`(i)` a black king
`(ii)` either a black card or a king
`(iii)` black and a king
`(iv)` a jack, queen or a king
(v) neither a hear nor king
(vi) spade or an ace
(vii) neither an ace nor a king
(viii) a diamond card
(ix) not a diamond card
(x) a black card

Answer

To find the probabilities for each part of the question regarding drawing a card from a standard deck of 52 cards, we will follow these steps:

### Total Outcomes
The total number of outcomes when drawing a card from a standard deck is always 52, as there are 52 cards in total.

### (i) Probability of drawing a black king
- **Favorable Outcomes**: There are 2 black kings (King of Spades and King of Clubs).
- **Probability**:
\[ P(\text{Black King}) = \frac{\text{Number of Black Kings}}{\text{Total Cards}} = \frac{2}{52} = \frac{1}{26} \]

### (ii) Probability of drawing either a black card or a king
- **Favorable Outcomes**: There are 26 black cards (13 Spades + 13 Clubs) and 4 kings (2 of which are black).
- To avoid double counting the black kings, we calculate:
\[ \text{Total Favorable Outcomes} = 26 + 4 - 2 = 28 \]
- **Probability**:
\[ P(\text{Black Card or King}) = \frac{28}{52} = \frac{7}{13} \]

### (iii) Probability of drawing a black card and a king
- **Favorable Outcomes**: There are 2 cards that are both black and kings (King of Spades and King of Clubs).
- **Probability**:
\[ P(\text{Black and King}) = \frac{2}{52} = \frac{1}{26} \]

### (iv) Probability of drawing a jack, queen, or king
- **Favorable Outcomes**: There are 4 jacks, 4 queens, and 4 kings.
- **Total Favorable Outcomes**:
\[ 4 + 4 + 4 = 12 \]
- **Probability**:
\[ P(\text{Jack, Queen, or King}) = \frac{12}{52} = \frac{3}{13} \]

### (v) Probability of drawing neither a heart nor a king
- **Favorable Outcomes**: There are 13 hearts and 4 kings (1 king of hearts).
- **Total Cards to Exclude**:
\[ 13 + 3 = 16 \text{ (since we exclude the King of Hearts)} \]
- **Remaining Cards**:
\[ 52 - 16 = 36 \]
- **Probability**:
\[ P(\text{Neither Heart nor King}) = \frac{36}{52} = \frac{9}{13} \]

### (vi) Probability of drawing a spade or an ace
- **Favorable Outcomes**: There are 13 spades and 4 aces (1 ace of spades).
- **Total Favorable Outcomes**:
\[ 13 + 4 - 1 = 16 \]
- **Probability**:
\[ P(\text{Spade or Ace}) = \frac{16}{52} = \frac{4}{13} \]

### (vii) Probability of drawing neither an ace nor a king
- **Favorable Outcomes**: There are 4 aces and 4 kings.
- **Total Cards to Exclude**:
\[ 4 + 4 = 8 \]
- **Remaining Cards**:
\[ 52 - 8 = 44 \]
- **Probability**:
\[ P(\text{Neither Ace nor King}) = \frac{44}{52} = \frac{11}{13} \]

### (viii) Probability of drawing a diamond card
- **Favorable Outcomes**: There are 13 diamond cards.
- **Probability**:
\[ P(\text{Diamond}) = \frac{13}{52} = \frac{1}{4} \]

### (ix) Probability of not drawing a diamond card
- **Favorable Outcomes**: There are 39 cards that are not diamonds (52 - 13).
- **Probability**:
\[ P(\text{Not Diamond}) = \frac{39}{52} = \frac{3}{4} \]

### (x) Probability of drawing a black card
- **Favorable Outcomes**: There are 26 black cards.
- **Probability**:
\[ P(\text{Black Card}) = \frac{26}{52} = \frac{1}{2} \]

### Summary of Probabilities
1. Black King: \( \frac{1}{26} \)
2. Black Card or King: \( \frac{7}{13} \)
3. Black and King: \( \frac{1}{26} \)
4. Jack, Queen, or King: \( \frac{3}{13} \)
5. Neither Heart nor King: \( \frac{9}{13} \)
6. Spade or Ace: \( \frac{4}{13} \)
7. Neither Ace nor King: \( \frac{11}{13} \)
8. Diamond Card: \( \frac{1}{4} \)
9. Not a Diamond Card: \( \frac{3}{4} \)
10. Black Card: \( \frac{1}{2} \)