One card is drawn from a pack of 52 cards, each of the 52 cards being equally likely to be drawn. Find the probability that the card drawn is:
(i) an ace.
(ii) red.
(iii) either red or king.
(iv) red and a king.
(v) a face card.
(vi) a red face card.
(vii) '2' of spades.
(viii) '10' of a black suit.

Answer

To solve the given problem step by step, we will calculate the probability for each part of the question based on the total number of cards in a standard deck (52 cards) and the number of favorable outcomes for each scenario.

### Step-by-Step Solution:

1. **Probability of drawing an Ace:**
- Total Aces in a deck = 4 (Ace of hearts, Ace of diamonds, Ace of clubs, Ace of spades)
- Total cards = 52
- Probability = Number of favorable outcomes / Total outcomes = 4 / 52 = 1 / 13

2. **Probability of drawing a red card:**
- Total red cards in a deck = 26 (13 hearts + 13 diamonds)
- Total cards = 52
- Probability = 26 / 52 = 1 / 2

3. **Probability of drawing either a red card or a king:**
- Total red cards = 26
- Total kings = 4 (2 red kings + 2 black kings)
- Total favorable outcomes = 26 (red cards) + 4 (kings) - 2 (since we counted the 2 red kings twice)
- Total favorable outcomes = 28
- Probability = 28 / 52 = 7 / 13

4. **Probability of drawing a red king:**
- Total red kings = 2 (King of hearts, King of diamonds)
- Total cards = 52
- Probability = 2 / 52 = 1 / 26

5. **Probability of drawing a face card:**
- Total face cards in a deck = 12 (3 face cards per suit × 4 suits)
- Total cards = 52
- Probability = 12 / 52 = 3 / 13

6. **Probability of drawing a red face card:**
- Total red face cards = 6 (3 face cards in hearts + 3 face cards in diamonds)
- Total cards = 52
- Probability = 6 / 52 = 3 / 26

7. **Probability of drawing the '2' of spades:**
- Total '2' of spades = 1
- Total cards = 52
- Probability = 1 / 52

8. **Probability of drawing '10' of a black suit:**
- Total black suits = 26 (13 clubs + 13 spades)
- Total '10's in black suits = 2 (10 of clubs, 10 of spades)
- Total cards = 52
- Probability = 2 / 52 = 1 / 26

### Summary of Probabilities:
- (i) Probability of Ace = 1/13
- (ii) Probability of Red Card = 1/2
- (iii) Probability of Either Red or King = 7/13
- (iv) Probability of Red and King = 1/26
- (v) Probability of Face Card = 3/13
- (vi) Probability of Red Face Card = 3/26
- (vii) Probability of '2' of Spades = 1/52
- (viii) Probability of '10' of a Black Suit = 1/26