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gmatrant
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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, for each value from 1 to 6, there are two cards in the deck with that 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?
a.8/33
b.62/165
c.17/33
d.103/165
e.25/33
My approach-
We need to calculate 1-p(x) where x is the probability of not drawing identical numbered cards.
1.From 12 cards we can pick any one number as the first card - No of ways =12
2.We now have 10 ways of picking the next card that does not have identical number = 10 ways
3. Picking up the 3rd card in 8 ways since 2 cards are already chosen and we cannot choose the same numbered card.
4. Similarly as in 3 we can now have 6 ways to choose the fourth card
Total ways of not picking even one identical number pair card = 12*10*8*6
Total ways of picking up 4 cards out of 12 = 12 * 11 * 10 *9
Probability of x is (12*10*8*6) / (12*11*10*9) = 16/33
Hence p(1-x) = 17/33.
I have a question here The total way of picking up 4 cards our of 12 is 12C4 and it will give me a different number than I calculated above. Can you please tell me where I am going wrong here, though I get the answer.
OA is C
a.8/33
b.62/165
c.17/33
d.103/165
e.25/33
My approach-
We need to calculate 1-p(x) where x is the probability of not drawing identical numbered cards.
1.From 12 cards we can pick any one number as the first card - No of ways =12
2.We now have 10 ways of picking the next card that does not have identical number = 10 ways
3. Picking up the 3rd card in 8 ways since 2 cards are already chosen and we cannot choose the same numbered card.
4. Similarly as in 3 we can now have 6 ways to choose the fourth card
Total ways of not picking even one identical number pair card = 12*10*8*6
Total ways of picking up 4 cards out of 12 = 12 * 11 * 10 *9
Probability of x is (12*10*8*6) / (12*11*10*9) = 16/33
Hence p(1-x) = 17/33.
I have a question here The total way of picking up 4 cards our of 12 is 12C4 and it will give me a different number than I calculated above. Can you please tell me where I am going wrong here, though I get the answer.
OA is C
