Question:
Case Study - 2: A group of friends wanted to play cards with two identical packs together. While shuffling the cards, three cards are dropped. Rest of the cards are shuffled and one card is drawn at random. Assuming that the dropped cards were a queen of hearts, a ten of spades and an ace of clubs, answer the following questions :
37(i)
Find the probability that the drawn card is a face card.
Case Study - 2: A group of friends wanted to play cards with two identical packs together. While shuffling the cards, three cards are dropped. Rest of the cards are shuffled and one card is drawn at random. Assuming that the dropped cards were a queen of hearts, a ten of spades and an ace of clubs, answer the following questions :
37(i) Find the probability that the drawn card is a face card.
37(i) Find the probability that the drawn card is a face card.
Show Hint
Always remember that an Ace is NOT a face card! Only Jacks, Queens, and Kings are classified as face cards.
This is a very common source of errors in card probability questions.
This is a very common source of errors in card probability questions.
Updated On: Jul 7, 2026
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Solution and Explanation
Step 1: Understanding the Question:
This question is from "Probability".
We are playing with two identical decks of 52 cards each.
Total cards in two packs $= 2 \times 52 = 104$.
Three cards are dropped: a queen of hearts, a ten of spades, and an ace of clubs.
Total remaining cards $= 104 - 3 = 101$. This represents our total number of possible outcomes.
We need to find the probability that a card drawn at random from the remaining cards is a face card.
Step 2: Key Formula or Approach:
1. Find the total number of face cards in two standard decks.
2. Subtract any face cards that were dropped to find the number of remaining face cards.
3. Apply the probability formula:
\[ P(\text{Face Card}) = \frac{\text{Remaining Face Cards}}{\text{Total Remaining Cards}} \]
Step 3: Detailed Explanation:
• Calculate the total number of face cards (Jacks, Queens, Kings) in two standard packs:
A single standard pack has $12$ face cards ($4$ Jacks, $4$ Queens, $4$ Kings).
Therefore, two identical packs have:
\[ 2 \times 12 = 24 \text{ face cards} \]
• Determine which of the dropped cards are face cards:
The dropped cards are:
- Queen of hearts (Face Card)
- Ten of spades (Numbered Card)
- Ace of clubs (Numbered/Letter Card, not a face card)
Only 1 face card (queen of hearts) was dropped.
• Calculate the remaining number of face cards in the combined pack:
\[ 24 - 1 = 23 \text{ face cards} \]
• Calculate the probability of drawing a face card:
\[ P(\text{Face Card}) = \frac{23}{101} \]
Step 4: Final Answer:
The probability that the drawn card is a face card is $\frac{23}{101}$.
This question is from "Probability".
We are playing with two identical decks of 52 cards each.
Total cards in two packs $= 2 \times 52 = 104$.
Three cards are dropped: a queen of hearts, a ten of spades, and an ace of clubs.
Total remaining cards $= 104 - 3 = 101$. This represents our total number of possible outcomes.
We need to find the probability that a card drawn at random from the remaining cards is a face card.
Step 2: Key Formula or Approach:
1. Find the total number of face cards in two standard decks.
2. Subtract any face cards that were dropped to find the number of remaining face cards.
3. Apply the probability formula:
\[ P(\text{Face Card}) = \frac{\text{Remaining Face Cards}}{\text{Total Remaining Cards}} \]
Step 3: Detailed Explanation:
• Calculate the total number of face cards (Jacks, Queens, Kings) in two standard packs:
A single standard pack has $12$ face cards ($4$ Jacks, $4$ Queens, $4$ Kings).
Therefore, two identical packs have:
\[ 2 \times 12 = 24 \text{ face cards} \]
• Determine which of the dropped cards are face cards:
The dropped cards are:
- Queen of hearts (Face Card)
- Ten of spades (Numbered Card)
- Ace of clubs (Numbered/Letter Card, not a face card)
Only 1 face card (queen of hearts) was dropped.
• Calculate the remaining number of face cards in the combined pack:
\[ 24 - 1 = 23 \text{ face cards} \]
• Calculate the probability of drawing a face card:
\[ P(\text{Face Card}) = \frac{23}{101} \]
Step 4: Final Answer:
The probability that the drawn card is a face card is $\frac{23}{101}$.
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