A card is drawn at random from a well shuffled pack of 52 cards. Find the probability that card drawn is:
(i) an ace
(ii) a heart
(iii) black
(iv) either an ace or black.

Explanation

To solve the problem step by step, we will calculate the probability for each part of the question regarding a card drawn from a standard deck of 52 cards.

### Step 1: Understanding the Deck of Cards
A standard deck of cards consists of 52 cards, which are divided into 4 suits:
- Hearts (♥)
- Diamonds (♦)
- Clubs (♣)
- Spades (♠)

Each suit has 13 cards, and there are 2 colors:
- Red: Hearts and Diamonds (26 cards)
- Black: Clubs and Spades (26 cards)

### Step 2: Calculate the Probability of Drawing an Ace
**(i) Probability of drawing an Ace:**
- There are 4 Aces in the deck (one from each suit).
- Total number of cards = 52.

**Probability (P) of drawing an Ace:**
\[ P(\text{Ace}) = \frac{\text{Number of Aces}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13} \]

### Step 3: Calculate the Probability of Drawing a Heart
**(ii) Probability of drawing a Heart:**
- There are 13 Hearts in the deck.
- Total number of cards = 52.

**Probability (P) of drawing a Heart:**
\[ P(\text{Heart}) = \frac{\text{Number of Hearts}}{\text{Total number of cards}} = \frac{13}{52} = \frac{1}{4} \]

### Step 4: Calculate the Probability of Drawing a Black Card
**(iii) Probability of drawing a Black card:**
- There are 26 Black cards (13 Clubs and 13 Spades) in the deck.
- Total number of cards = 52.

**Probability (P) of drawing a Black card:**
\[ P(\text{Black}) = \frac{\text{Number of Black cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2} \]

### Step 5: Calculate the Probability of Drawing Either an Ace or a Black Card
**(iv) Probability of drawing either an Ace or a Black card:**
- Total number of Black cards = 26.
- Number of Aces = 4 (1 Ace from each suit).
- Among the Black cards, there are 2 Aces (Ace of Spades and Ace of Clubs).

Using the principle of inclusion-exclusion:
\[ P(\text{Ace or Black}) = P(\text{Ace}) + P(\text{Black}) - P(\text{Ace and Black}) \]
Where:
- \(P(\text{Ace}) = \frac{4}{52}\)
- \(P(\text{Black}) = \frac{26}{52}\)
- \(P(\text{Ace and Black}) = \frac{2}{52}\) (since there are 2 black Aces)

Calculating:
\[ P(\text{Ace or Black}) = \frac{4}{52} + \frac{26}{52} - \frac{2}{52} = \frac{4 + 26 - 2}{52} = \frac{28}{52} = \frac{7}{13} \]

### Summary of Probabilities
1. Probability of drawing an Ace: \(\frac{1}{13}\)
2. Probability of drawing a Heart: \(\frac{1}{4}\)
3. Probability of drawing a Black card: \(\frac{1}{2}\)
4. Probability of drawing either an Ace or a Black card: \(\frac{7}{13}\)