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Probability Question- Cards pick up

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by Saurabh211 » Mon Jun 24, 2013 11:17 am
Bill has small deck of 12 cards, made up of 2 suites of 6 cards each valued 1 to 6. Thus there are two cards in the deck with same value. He draws 4 cards one by one without replacement and looks for pair of cards that have same value. What is the probability that he gets
a) exactly one pair
b) exactly two pairs
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by srcc25anu » Mon Jun 24, 2013 4:05 pm
# ways to draw 4 cards out of 12 (no restriction) = 12 * 11 * 10 * 9
# ways to draw 4 DIFFERENT value cards out of 12 cards = 12 * 10 * 8 * 6
Prob to draw 4 cards that no cards are same = 12 * 10 * 8 * 6 / 12 * 11 * 10 * 9
= 16/33

Part A
# ways to draw 4 cards out of 12 (no restriction) = 12 * 11 * 10 * 9
# ways to draw 4 cards such that we have EXACTLY 1 PAIR = 12 * 1 * 10 * 8
Now EIther 1st or 2nd cards could be same, or 1st and 3rd cards could be same. So Number of such arrangements of 4 cards where 2 are same = 4C2 = 6
Prob to draw 4 cards that EXACTLY 2 cards are same (1 PAIR and 2 different cards) = 12 * 1 * 10 * 8 * 4C2 / 12 * 11 * 10 * 9
= 16/33

Part B
# ways to draw 4 cards out of 12 (no restriction) = 12 * 11 * 10 * 9
# ways to draw 4 cards such that we have EXACTLY 2 PAIRS = 12 * 1 * 10 * 1
No of arrangements = 4C2 / 2 = 3 (since AABB and BBAA are in fact the same arrangement for this purpose where 1st card = 2nd card and 3rd card = 4th card)
Prob to draw 4 cards that we get EXACTLY 2 PAIRS = 12 * 1 * 10 * 1 * (4C2/2)/ 12 * 11 * 10 * 9
= 1/33
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