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From an ordinary deck of cards, only 12 picture cards are [#permalink]

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19 Apr 2006, 14:09

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From an ordinary deck of cards, only 12 picture cards are retained. They are shuffled (mixed) and a man draws 2 cards at random. He then announce one of these cards is the King or Spades. What is the probability that he holds two Kings? (hint: list the possible sample space carefully).

Consider the table below (I am not good at cards)
.............Spades......Hearts.........Something........Something
King............1............1..................1.......................1
Queen.........2............2..................2.......................2
Jack............3............3..................3.......................3

In the 12 picture cards, We have a spade and a king (and it could be a spade king too).

Spade (Non King) X Any (Non Spade Non King)= 2 x 6 = 12
King (Non Spade) x Any (Non King Non Spade) = 3 x 6 = 18
Spade King x (Any Non Spade Non King) = 1 x 6 = 6
Spade King x Any Non Spade King = 1 x 3
Non Spade King x Non King Spade = 3x2 = 6.
Thus total sample size is 45.
Out of this, Two kings are C(4, 2) = 12.
Thus probability = 12/45 = 4/15.
_________________

Is it 1/35? I used conditional probability and got this value...
_________________

"To dream anything that you want to dream, that is the beauty of the human mind. To do anything that you want to do, that is the strength of the human will. To trust yourself, to test your limits, that is the courage to succeed."

From an ordinary deck of cards, only 12 picture cards are retained. They are shuffled (mixed) and a man draws 2 cards at random. He then announce one of these cards is the King or Spades. What is the probability that he holds two Kings? (hint: list the possible sample space carefully).

Knowing that it is a King or Spade we have 3 sets of possibilities for the 2 cards:
1. (King, Spade) = 4+4+4+3 = 15
2. (Spade, Spade) = 6
3. (King, King) = 6

K of spade is part of all three of thse possibilities.
Number of overlaps in #2 and #3 is exactly 3*2 =6

Total = 15+6+6-3*2 = 21

2 kings can occur in 4C2 = 6 ways
Hence P(2 Kings) = 6/21 = 2/7
_________________

"To dream anything that you want to dream, that is the beauty of the human mind. To do anything that you want to do, that is the strength of the human will. To trust yourself, to test your limits, that is the courage to succeed."

I think it is not that hard. Since the player have announced that the first
card is a king we should not account for it is not. So the probability
that the player has two kings is a probability that the second card is a king. And this leaves us with 3/11.

I think it is not that hard. Since the player have announced that the first card is a king we should not account for it is not. So the probability that the player has two kings is a probability that the second card is a king. And this leaves us with 3/11.

What is OA ?

OA is 3/11. And the official explanation is same as above only. But I am also not fully convinced by OE.

Because one of the card can be either a king or a spade or both. Note that amongst the 12 picture cards, there are 4 kings and 3 spades. But the 3 spades also include the spades King.

So one of the two cards probability to be king is 4/6. And the other card's probability is 3/11.

From an ordinary deck of cards, only 12 picture cards are retained. They are shuffled (mixed) and a man draws 2 cards at random. He then announce one of these cards is the King or Spades. What is the probability that he holds two Kings? (hint: list the possible sample space carefully).

It should be King "of" spades....i think there is a typo in the question

I think it is not that hard. Since the player have announced that the first card is a king we should not account for it is not. So the probability that the player has two kings is a probability that the second card is a king. And this leaves us with 3/11.

What is OA ?

OA is 3/11. And the official explanation is same as above only. But I am also not fully convinced by OE.

Because one of the card can be either a king or a spade or both. Note that amongst the 12 picture cards, there are 4 kings and 3 spades. But the 3 spades also include the spades King.

So one of the two cards probability to be king is 4/6. And the other card's probability is 3/11.

=> 4/6 * 3/11 * 2C1 = 4/11.

Pls feel free to discuss this more.

Thanks, Viipin

Vipin, IMHO if you give us the OE as Deowl explained (he explained as if the one of the card is the King of Spades, but it's not stated in the question) there is a typo in the question in third sentence. "... He then announce one of these cards is the King OR Spades..."
Accrding to you and to Deowl there must be 'OF'

Am I right?

In any case let's discuss it further from the point that he one of the card is the King OR Spades.

I think it is not that hard. Since the player have announced that the first card is a king we should not account for it is not. So the probability that the player has two kings is a probability that the second card is a king. And this leaves us with 3/11.

What is OA ?

OA is 3/11. And the official explanation is same as above only. But I am also not fully convinced by OE.

Because one of the card can be either a king or a spade or both. Note that amongst the 12 picture cards, there are 4 kings and 3 spades. But the 3 spades also include the spades King.

So one of the two cards probability to be king is 4/6. And the other card's probability is 3/11.

=> 4/6 * 3/11 * 2C1 = 4/11.

Pls feel free to discuss this more.

Thanks, Viipin

Vipin, IMHO if you give us the OE as Deowl explained (he explained as if the one of the card is the King of Spades, but it's not stated in the question) there is a typo in the question in third sentence. "... He then announce one of these cards is the King OR Spades..." Accrding to you and to Deowl there must be 'OF'

Am I right?

In any case let's discuss it further from the point that he one of the card is the King OR Spades.

Hi M8,
Actually, I didn't even realize that the original question had a typo untill Shevy mentioned it.
I think it has become a little confusing. I will try to sort things out here.

Case1: The man anounced that one of the cards is a king OF spades.

Off. Answer to case1 is 3/11.
Since we already know that one of the cards is a king, the other card can be chosen out of the remaining 3 king cards. so the probability is -

3 / 11.

Case2: The man anounced that one of the cards is a king OR spades.

No OA answer for this since this was not the original problem. My answer to this would be 4/11.

Since, one of the card can be either a king or a spade or both. Note that amongst the 12 picture cards, there are 4 kings and 3 spades. But the 3 spades also include the King of Spades.

So one of the two cards probability to be king is 4/6.