Two cards are drawn at random from a pack of 52 cards. The probability of getting at least a spade and an ace is

Options

A. `1//34`
B. `8//221`
C. `29//442`
D. `2//51`

Answer

To find the probability of drawing at least one spade and one ace when two cards are drawn from a standard deck of 52 cards, we can follow these steps:

### Step 1: Identify the total number of cards and relevant categories
A standard deck has 52 cards, which includes:
- 13 Spades (including the Ace of Spades)
- 4 Aces (one of each suit: Spades, Hearts, Diamonds, Clubs)

### Step 2: Define the event of interest
We want to find the probability of drawing at least one spade and one ace. This can happen in two scenarios:
1. One card is a spade (which could be the Ace of Spades) and the other card is an Ace (which could be of any suit).
2. Both cards are spades, and one of them is an Ace.

### Step 3: Calculate the number of favorable outcomes
1. **Case 1:** One spade and one ace (Ace can be of any suit).
- Choose 1 Ace from the 4 Aces: \( \binom{4}{1} = 4 \)
- Choose 1 Spade from the 12 remaining Spades (excluding the Ace of Spades): \( \binom{12}{1} = 12 \)
- Total ways for this case: \( 4 \times 12 = 48 \)

2. **Case 2:** One Ace of Spades and one other Spade.
- Choose the Ace of Spades: \( \binom{1}{1} = 1 \)
- Choose 1 from the remaining 12 Spades: \( \binom{12}{1} = 12 \)
- Total ways for this case: \( 1 \times 12 = 12 \)

### Step 4: Combine the cases
The total number of favorable outcomes is the sum of the two cases:
- Total favorable outcomes = Case 1 + Case 2 = \( 48 + 12 = 60 \)

### Step 5: Calculate the total number of ways to choose 2 cards from 52
The total number of ways to choose 2 cards from 52 is given by:
\[ \binom{52}{2} = \frac{52 \times 51}{2} = 1326 \]

### Step 6: Calculate the probability
The probability of drawing at least one spade and one ace is given by:
\[ P(\text{at least one spade and one ace}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{60}{1326} \]

### Step 7: Simplify the probability
To simplify \( \frac{60}{1326} \):
- The GCD of 60 and 1326 is 6.
- Dividing both by 6 gives:
\[ \frac{60 \div 6}{1326 \div 6} = \frac{10}{221} \]

### Final Answer
The probability of drawing at least one spade and one ace when two cards are drawn from a pack of 52 cards is:
\[ \frac{10}{221} \]