cards pick probabilty

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mikoamar

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Jun 14, 2019
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hey i have a question i need help with:
4 cards taken out of a standard card pack (52 crads) without returning
what is the probabilty to take out exactly one pair?

i tried to pick the pair first and then pick the cards out of the specific pair but couldnt figure out how to continue
1573927752051.png
 
What instruction do you have to follow to get the pair from picking 4 cards.
1) Pick the denomination which is 13C1 (you did this)
2) Pick the 2 cards from the 4 of the selected denomination which is 4C2 (you did this)
What are the rest of the steps/decision?

You are on a good roll, please keep trying and report back.
 
It might not be as straightforward as it looks...

Break the deck up into 13 exclusive sets that are each rank.
You choose 2 from one of these sets, and 1 each from 2 of them.
Choose the set you pick the pair from. Choose the 2 sets you pick a singleton from.
Then choose 2 cards from the 4 of the pair rank, and 1 card each from the 4 of the 2 singleton ranks.
Divide all of this by the total number of ways to select 4 cards from the 52.

\(\displaystyle p =\dfrac{ \dbinom{13}{1}\dbinom{12}{2}\dbinom{4}{2}\dbinom{4}{1}\dbinom{4}{1}}{\dbinom{52}{4}} = \dfrac{6336}{20825}\)
 
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