One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally likely, calculate the probability that the card will be
(i) a diamond (ii) not an ace (iii) not a king (iv) a black card (i.e. a club or a spade)
(v) a red card
(vi) not a diamond (vii) not a black card.

Answer

To solve the problem step by step, we need to calculate the probability of drawing a specific type of card from a standard deck of 52 cards.

### Total Cards in a Deck:
- A standard deck has **52 cards**.

### Card Distribution:
- **Diamonds:** 13 cards
- **Hearts:** 13 cards
- **Clubs:** 13 cards
- **Spades:** 13 cards
- **Total Red Cards (Diamonds + Hearts):** 26 cards
- **Total Black Cards (Clubs + Spades):** 26 cards
- **Total Aces:** 4 cards
- **Total Kings:** 4 cards

### Probability Calculation:
The probability \( P \) of an event is given by the formula:
\[ P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

Now, we will calculate the probability for each case:

#### (i) Probability of drawing a diamond:
- Favorable outcomes = 13 (number of diamonds)
- Total outcomes = 52
\[ P(\text{diamond}) = \frac{13}{52} = \frac{1}{4} \]

#### (ii) Probability of drawing a card that is not an ace:
- Favorable outcomes = 52 - 4 = 48 (total cards - aces)
- Total outcomes = 52
\[ P(\text{not an ace}) = \frac{48}{52} = \frac{12}{13} \]

#### (iii) Probability of drawing a card that is not a king:
- Favorable outcomes = 52 - 4 = 48 (total cards - kings)
- Total outcomes = 52
\[ P(\text{not a king}) = \frac{48}{52} = \frac{12}{13} \]

#### (iv) Probability of drawing a black card (club or spade):
- Favorable outcomes = 26 (number of black cards)
- Total outcomes = 52
\[ P(\text{black card}) = \frac{26}{52} = \frac{1}{2} \]

#### (v) Probability of drawing a red card (heart or diamond):
- Favorable outcomes = 26 (number of red cards)
- Total outcomes = 52
\[ P(\text{red card}) = \frac{26}{52} = \frac{1}{2} \]

#### (vi) Probability of drawing a card that is not a diamond:
- Favorable outcomes = 52 - 13 = 39 (total cards - diamonds)
- Total outcomes = 52
\[ P(\text{not a diamond}) = \frac{39}{52} = \frac{3}{4} \]

#### (vii) Probability of drawing a card that is not a black card:
- Favorable outcomes = 52 - 26 = 26 (total cards - black cards)
- Total outcomes = 52
\[ P(\text{not a black card}) = \frac{26}{52} = \frac{1}{2} \]

### Summary of Probabilities:
1. Probability of diamond: \( \frac{1}{4} \)
2. Probability of not an ace: \( \frac{12}{13} \)
3. Probability of not a king: \( \frac{12}{13} \)
4. Probability of black card: \( \frac{1}{2} \)
5. Probability of red card: \( \frac{1}{2} \)
6. Probability of not a diamond: \( \frac{3}{4} \)
7. Probability of not a black card: \( \frac{1}{2} \)