A card is drawn from a well shuffled deck of 52 cards. Find the probability of drawing.
(i) a black king
(ii) a jack, queen, king or an ace.
(iii) a card, which is neither a heart nor a king
(iv) a spade or club.

Answer

To solve the problem, we will calculate the probability for each part step by step.

### Total Number of Cards
A standard deck of cards has a total of 52 cards.

### (i) Probability of drawing a black king
1. **Identify the favorable outcomes**: There are 2 black kings in a deck (the King of Spades and the King of Clubs).
2. **Total outcomes**: The total number of cards is 52.
3. **Calculate the probability**:
\[ P(\text{black king}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{52} = \frac{1}{26} \]

### (ii) Probability of drawing a jack, queen, king, or an ace
1. **Identify the favorable outcomes**: There are 4 jacks, 4 queens, 4 kings, and 4 aces in the deck.
- Total favorable outcomes = \(4 + 4 + 4 + 4 = 16\)
2. **Total outcomes**: The total number of cards is 52.
3. **Calculate the probability**:
\[ P(\text{jack, queen, king, or ace}) = \frac{16}{52} = \frac{4}{13} \]

### (iii) Probability of drawing a card which is neither a heart nor a king
1. **Identify the unfavorable outcomes**:
- Total hearts = 13
- Total kings = 4 (but 1 of these is a heart, so we only count 3 additional kings)
- Total unfavorable outcomes = \(13 + 3 = 16\)
2. **Total outcomes**: The total number of cards is 52.
3. **Calculate the favorable outcomes**:
\[ \text{Favorable outcomes} = \text{Total outcomes} - \text{Unfavorable outcomes} = 52 - 16 = 36 \]
4. **Calculate the probability**:
\[ P(\text{neither heart nor king}) = \frac{36}{52} = \frac{9}{13} \]

### (iv) Probability of drawing a spade or club
1. **Identify the favorable outcomes**: There are 13 spades and 13 clubs.
- Total favorable outcomes = \(13 + 13 = 26\)
2. **Total outcomes**: The total number of cards is 52.
3. **Calculate the probability**:
\[ P(\text{spade or club}) = \frac{26}{52} = \frac{1}{2} \]

### Summary of Results
- (i) Probability of drawing a black king: \(\frac{1}{26}\)
- (ii) Probability of drawing a jack, queen, king, or an ace: \(\frac{4}{13}\)
- (iii) Probability of drawing a card which is neither a heart nor a king: \(\frac{9}{13}\)
- (iv) Probability of drawing a spade or club: \(\frac{1}{2}\)