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# Bill has a small deck of 12 playing cards made up of only 2

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Manager
Joined: 19 Aug 2007
Posts: 170
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Kudos [?]: 14 [0], given: 0

Bill has a small deck of 12 playing cards made up of only 2 [#permalink]  20 Oct 2007, 12:16
Bill has a small deck of 12 playing cards made up of only 2 suits of 6 cards each. Each of the 6 cards within a suit has a different value from 1 to 6; thus, there are 2 cards in the deck that have the same value.

Bill likes to play a game in which he shuffles the deck, turns over 4 cards, and looks for pairs of cards that have the same value. What is the chance that Bill finds at least one pair of cards that have the same value?

8/33

62/165

17/33

103/165

25/33
VP
Joined: 08 Jun 2005
Posts: 1147
Followers: 6

Kudos [?]: 127 [0], given: 0

I'm getting 17/33

1*10/11*8/10*6/9 = 48/99

1-48/99 = 1-16/33 = 17/33

1 ---> choosing any card

10/11 ---> not choosing his pair out of 11 cards.

8/10 ---> now you have two cards. not choosing their pairs.

6/9 ---> you have choosen three cards ! not choosing their pairs.

Intern
Joined: 27 Jul 2007
Posts: 38
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Kudos [?]: 4 [0], given: 0

nice explanation killersquirrel.

im convinced of 17/33 as well
CEO
Joined: 21 Jan 2007
Posts: 2766
Location: New York City
Followers: 9

Kudos [?]: 332 [0], given: 4

KillerSquirrel wrote:
I'm getting 17/33

1*10/11*8/10*6/9 = 48/99

1-48/99 = 1-16/33 = 17/33

1 ---> choosing any card

10/11 ---> not choosing his pair out of 11 cards.

8/10 ---> now you have two cards. not choosing their pairs.

6/9 ---> you have choosen three cards ! not choosing their pairs.

How would your answer change it it said at least TWO pairs will the same value?
VP
Joined: 08 Jun 2005
Posts: 1147
Followers: 6

Kudos [?]: 127 [0], given: 0

bmwhype2 wrote:
KillerSquirrel wrote:
I'm getting 17/33

1*10/11*8/10*6/9 = 48/99

1-48/99 = 1-16/33 = 17/33

1 ---> choosing any card

10/11 ---> not choosing his pair out of 11 cards.

8/10 ---> now you have two cards. not choosing their pairs.

6/9 ---> you have choosen three cards ! not choosing their pairs.

How would your answer change it it said at least TWO pairs will the same value?

Since you have four cards the most you can have is two pairs (can never get three pairs in four cards !!) so you really ask for "what is the probability for him to choose two pairs" (i.e 2,2 and 4,4).

(1*1/11*1*1/9) + (1*10/11*2/10*1/9) + (1*1/11*1*1/9) = 1/99 + 2/99 + 1/99 = 4/99

Manager
Joined: 29 Jul 2007
Posts: 182
Followers: 1

Kudos [?]: 13 [0], given: 0

KillerSquirrel wrote:
bmwhype2 wrote:
KillerSquirrel wrote:
I'm getting 17/33

1*10/11*8/10*6/9 = 48/99

1-48/99 = 1-16/33 = 17/33

1 ---> choosing any card

10/11 ---> not choosing his pair out of 11 cards.

8/10 ---> now you have two cards. not choosing their pairs.

6/9 ---> you have choosen three cards ! not choosing their pairs.

How would your answer change it it said at least TWO pairs will the same value?

Since you have four cards the most you can have is two pairs (can never get three pairs in four cards !!) so you really ask for "what is the probability for him to choose two pairs" (i.e 2,2 and 4,4).

(1*1/11*1*1/9) + (1*10/11*2/10*1/9) + (1*1/11*1*1/9) = 1/99 + 2/99 + 1/99 = 4/99

Why the 2nd ((1*1/11*1*1/9))?
VP
Joined: 08 Jun 2005
Posts: 1147
Followers: 6

Kudos [?]: 127 [0], given: 0

Skewed wrote:
KillerSquirrel wrote:
bmwhype2 wrote:
KillerSquirrel wrote:
I'm getting 17/33

1*10/11*8/10*6/9 = 48/99

1-48/99 = 1-16/33 = 17/33

1 ---> choosing any card

10/11 ---> not choosing his pair out of 11 cards.

8/10 ---> now you have two cards. not choosing their pairs.

6/9 ---> you have choosen three cards ! not choosing their pairs.

How would your answer change it it said at least TWO pairs will the same value?

Since you have four cards the most you can have is two pairs (can never get three pairs in four cards !!) so you really ask for "what is the probability for him to choose two pairs" (i.e 2,2 and 4,4).

(1*1/11*1*1/9) + (1*10/11*2/10*1/9) + (1*1/11*1*1/9) = 1/99 + 2/99 + 1/99 = 4/99

Why the 2nd ((1*1/11*1*1/9))?

Assume

2 - 2 - 4 - 4

or

2 - 4 - 4 - 2

or

4 - 4 - 2 - 2

Manager
Joined: 29 Jul 2007
Posts: 182
Followers: 1

Kudos [?]: 13 [0], given: 0

But there are 6 variations of 2 2's and 2 4's.
2244
2424
2442
4422
4242
4224

4!/(2!2!)=6

Why are only 3 listed?
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