Two cards are drawn without replacement from a well-suffled pack of 52 carbs. The probability that one is spade and the other is queen of red colour is

Options

A. `(2)/(51)`
B. `(1)/(7)`
C. `(1)/(51)`
D. `(5)/(51)`

Answer

To solve the problem of finding the probability that one card drawn is a spade and the other is a queen of red color from a standard deck of 52 cards, we can follow these steps:

### Step 1: Understand the Deck Composition
A standard deck has:
- 52 cards in total
- 13 spades (black)
- 13 clubs (black)
- 13 diamonds (red)
- 13 hearts (red)

Among the red cards (diamonds and hearts), there are 2 queens: the Queen of Diamonds and the Queen of Hearts.

### Step 2: Define the Event
We need to find the probability of drawing:
- One spade
- One queen of red color (which can be either the Queen of Diamonds or the Queen of Hearts)

### Step 3: Calculate the Total Number of Ways to Draw Two Cards
The total number of ways to draw 2 cards from 52 cards is given by the combination formula:
\[ n(S) = \binom{52}{2} = \frac{52 \times 51}{2} = 1326 \]

### Step 4: Calculate the Number of Favorable Outcomes
We can have two scenarios for drawing one spade and one red queen:
1. Draw a spade first and then a red queen.
2. Draw a red queen first and then a spade.

#### Scenario 1: Spade first, then red queen
- The number of ways to choose 1 spade from 13 spades: \(\binom{13}{1} = 13\)
- The number of ways to choose 1 red queen from 2 red queens: \(\binom{2}{1} = 2\)

Thus, the total ways for this scenario:
\[ n(E_1) = 13 \times 2 = 26 \]

#### Scenario 2: Red queen first, then spade
- The number of ways to choose 1 red queen from 2 red queens: \(\binom{2}{1} = 2\)
- The number of ways to choose 1 spade from 13 spades: \(\binom{13}{1} = 13\)

Thus, the total ways for this scenario:
\[ n(E_2) = 2 \times 13 = 26 \]

### Step 5: Total Favorable Outcomes
Since both scenarios are mutually exclusive, we can add the outcomes:
\[ n(E) = n(E_1) + n(E_2) = 26 + 26 = 52 \]

### Step 6: Calculate the Probability
The probability \(P\) that one card is a spade and the other is a queen of red color is given by:
\[ P = \frac{n(E)}{n(S)} = \frac{52}{1326} \]

### Step 7: Simplify the Probability
We can simplify this fraction:
\[ P = \frac{52}{1326} = \frac{1}{51} \]

### Final Answer
Thus, the probability that one card is a spade and the other is a queen of red color is:
\[ \boxed{\frac{1}{51}} \]