Question

All the red face cards are removed from a pack of 52 playing cards. The remaining cards are well shuffled and then a card is drawn at random. Find the probability of getting a : (i) red card (ii) a king or queen.

• A. (i) \(\frac{10}{23}\), (ii) \(\frac{2}{23}\)
• B. (i) \(\frac{1}{2}\), (ii) \(\frac{1}{13}\)
• C. (i) \(\frac{11}{23}\), (ii) \(\frac{4}{23}\)
• D. (i) \(\frac{10}{23}\), (ii) \(\frac{3}{23}\)

Hint:

Always calculate the new total number of outcomes first in conditional deck problems where cards are removed.
Standard cards calculations will lead to wrong options if the reduced denominator of 46 is not utilized.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
Initially, we have a standard deck of 52 playing cards. All the red face cards are removed. A single card is drawn from the remaining deck. We need to calculate:
1. The probability of getting a red card.
2. The probability of getting a king or a queen.

Step 2: Key Formula or Approach:
The probability of an event is given by:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

Step 3: Detailed Explanation:
1.

Determine the total number of remaining cards:
A standard deck has 52 cards.
Face cards are Kings, Queens, and Jacks. There are 3 face cards per suit.
The red suits are Hearts and Diamonds.
Red face cards = 3 (from Hearts) + 3 (from Diamonds) = 6 cards.
The total number of remaining cards in the deck is:
\[ \text{Total remaining cards} = 52 - 6 = 46 \]

2.

Part (i): Find the probability of getting a red card:
A standard deck has 26 red cards (13 Hearts, 13 Diamonds).
Since we removed the 6 red face cards, the number of remaining red cards is:
\[ \text{Remaining red cards} = 26 - 6 = 20 \]
The probability of drawing a red card is:
\[ P(\text{Red Card}) = \frac{20}{46} = \frac{10}{23} \]

3.

Part (ii): Find the probability of getting a king or queen:
A standard deck has 4 Kings and 4 Queens.
Among these, the red Kings (2) and red Queens (2) are face cards and have been removed.
Only the black Kings (2) and black Queens (2) remain in the deck.
Therefore, the number of remaining Kings or Queens is:
\[ 2 \text{ (black Kings)} + 2 \text{ (black Queens)} = 4 \]
The probability of drawing a king or queen is:
\[ P(\text{King or Queen}) = \frac{4}{46} = \frac{2}{23} \]

Step 4: Final Answer:
The probabilities are \(\frac{10}{23}\) for drawing a red card and \(\frac{2}{23}\) for drawing a king or queen, which corresponds to option (A).