From a well shuffeled deck of 52 cards, one card is drawn. Find the probability that the card drawn is (i) a face card (ii) not a face card (iii) a queen of black colour (iv) a card with nurnbcr 5 or 6 (v) a card with number less than 8 (vi) a card with number between 2 and 9.

Answer

To solve the problem, we will find the probability for each of the given scenarios step by step.

### Total Outcomes
In a standard deck of cards, there are 52 cards in total.

### (i) Probability of drawing a face card
- **Face cards**: There are 3 face cards (Jack, Queen, King) in each of the 4 suits (Hearts, Diamonds, Clubs, Spades).
- **Total face cards**: \(3 \times 4 = 12\)
- **Probability**:
\[ P(\text{face card}) = \frac{\text{Number of face cards}}{\text{Total cards}} = \frac{12}{52} = \frac{3}{13} \]

### (ii) Probability of not drawing a face card
- **Non-face cards**: Total cards - Face cards = \(52 - 12 = 40\)
- **Probability**:
\[ P(\text{not a face card}) = \frac{\text{Number of non-face cards}}{\text{Total cards}} = \frac{40}{52} = \frac{10}{13} \]

### (iii) Probability of drawing a queen of black color
- **Black queens**: There are 2 black queens (Queen of Spades and Queen of Clubs).
- **Probability**:
\[ P(\text{black queen}) = \frac{\text{Number of black queens}}{\text{Total cards}} = \frac{2}{52} = \frac{1}{26} \]

### (iv) Probability of drawing a card with number 5 or 6
- **Cards with number 5 or 6**: There are 2 cards (5 and 6) in each of the 4 suits.
- **Total cards with 5 or 6**: \(2 \times 4 = 8\)
- **Probability**:
\[ P(\text{5 or 6}) = \frac{\text{Number of cards with 5 or 6}}{\text{Total cards}} = \frac{8}{52} = \frac{2}{13} \]

### (v) Probability of drawing a card with number less than 8
- **Cards numbered less than 8**: The numbers are 2, 3, 4, 5, 6, and 7 (6 numbers).
- **Total cards with numbers less than 8**: \(6 \times 4 = 24\)
- **Probability**:
\[ P(\text{less than 8}) = \frac{\text{Number of cards less than 8}}{\text{Total cards}} = \frac{24}{52} = \frac{6}{13} \]

### (vi) Probability of drawing a card with number between 2 and 9
- **Cards numbered between 2 and 9**: The numbers are 2, 3, 4, 5, 6, 7, 8, and 9 (8 numbers).
- **Total cards with numbers between 2 and 9**: \(8 \times 4 = 32\)
- **Probability**:
\[ P(\text{between 2 and 9}) = \frac{\text{Number of cards between 2 and 9}}{\text{Total cards}} = \frac{32}{52} = \frac{8}{13} \]

### Summary of Probabilities
1. Probability of a face card: \( \frac{3}{13} \)
2. Probability of not a face card: \( \frac{10}{13} \)
3. Probability of a black queen: \( \frac{1}{26} \)
4. Probability of a card with number 5 or 6: \( \frac{2}{13} \)
5. Probability of a card with number less than 8: \( \frac{6}{13} \)
6. Probability of a card with number between 2 and 9: \( \frac{8}{13} \)