2 cards are selected randomly from a pack of 52 cards. The probability that the first card is a club and second is not a king, is

Options

A. `82/221`
B. `41/221`
C. `1/221`
D. `4/17`

Answer

To solve the problem of finding the probability that the first card drawn is a club and the second card drawn is not a king, we can break it down into steps.

### Step 1: Understand the total number of cards
A standard deck of cards has 52 cards, which consist of 4 suits: clubs, diamonds, hearts, and spades. Each suit has 13 cards.

**Hint:** Remember that there are 52 cards in total, divided equally among 4 suits.

### Step 2: Calculate the probability of drawing a club first
The number of clubs in a deck is 13. Therefore, the probability of drawing a club first is:

\[ P(\text{First card is a club}) = \frac{13}{52} = \frac{1}{4} \]

**Hint:** The probability of an event is the number of favorable outcomes divided by the total number of outcomes.

### Step 3: Calculate the probability of drawing a non-king second
After drawing the first card (which is a club), there are now 51 cards left in the deck. We need to consider two scenarios for the second card:

1. If the first card was a club but not a king (12 clubs are not kings).
2. If the first card was the king of clubs (1 club is a king).

**Case 1:** First card is a club but not a king.
- If the first card is one of the 12 non-king clubs, there are still 4 kings remaining among the 51 cards. Thus, the number of non-kings left is \(51 - 4 = 47\).
- The probability for this case is:

\[ P(\text{Second card is not a king | First card is a non-king club}) = \frac{47}{51} \]

**Case 2:** First card is the king of clubs.
- If the first card is the king of clubs, there are now 3 kings left among the 51 cards. Thus, the number of non-kings left is \(51 - 3 = 48\).
- The probability for this case is:

\[ P(\text{Second card is not a king | First card is the king of clubs}) = \frac{48}{51} \]

### Step 4: Combine the probabilities
Now, we can combine the probabilities from both cases using the law of total probability:

1. Probability of first card being a non-king club and second card not being a king:
\[ P(\text{Non-king club}) \times P(\text{Not a king | Non-king club}) = \frac{12}{52} \times \frac{47}{51} \]

2. Probability of first card being the king of clubs and second card not being a king:
\[ P(\text{King of clubs}) \times P(\text{Not a king | King of clubs}) = \frac{1}{52} \times \frac{48}{51} \]

### Step 5: Calculate the total probability
Now, we can calculate the total probability:

\[ P(\text{First card is a club and second is not a king}) = \left(\frac{12}{52} \times \frac{47}{51}\right) + \left(\frac{1}{52} \times \frac{48}{51}\right) \]

Calculating each term:

1. For the first term:
\[ \frac{12 \times 47}{52 \times 51} = \frac{564}{2652} \]

2. For the second term:
\[ \frac{1 \times 48}{52 \times 51} = \frac{48}{2652} \]

Now, adding these two fractions:

\[ \frac{564 + 48}{2652} = \frac{612}{2652} \]

### Step 6: Simplify the fraction
Now, we simplify \(\frac{612}{2652}\):

\[ \frac{612 \div 12}{2652 \div 12} = \frac{51}{221} \]

Thus, the final answer is:

\[ \text{Probability that the first card is a club and the second is not a king} = \frac{51}{221} \]

### Summary of Steps
1. Understand the total number of cards.
2. Calculate the probability of drawing a club first.
3. Calculate the probability of drawing a non-king second based on the first card drawn.
4. Combine the probabilities from both cases.
5. Calculate and simplify the total probability.