Where do I start? Statistics Question.

[Peachyyy]

Hi everyone,

I have a beginning statistics problem and I'm not even sure where to begin. I'm sure if I could get some help with where to start, I could get the answer.

In a poker game, five cards are dealt at random from an ordinary deck of 52 playing cards. Find the probabilities of getting two pairs (any two disinct face values occuring exactly twice).

Thanks in advance for any help you can offer.

 

[Peachyyy]

This is my instructors advice, but i'm not even sure I understand that :?

Getting two pairs among the 5 cards means we need two cards of the same value, two cards of another value, and a single card of a third value. For example, if you have 2 of club, 2 of heart, king of diamond, king of heart, and 7 of spade; that is, you have a pair of 2's and a pair of king's among the 5 cards in your hand, then you get two pairs.

Now you need to figure out how many ways you can get two pairs and a single card among a deck of 52 cards for this event. You will also need to find out how many ways to draw 5 cards from the deck for the sample space

 

[pka]

There are 13 denominations: There are C(13,2)=[13!]/{(2!)(11!)} ways to chose two of them.
There C(4,2)*C(4,2) ways to chose two cards from each of the two chosen denominations.
Then there are 44 ways to choose the fifth non-matching card.
There are C(52,5) ways to choose five cards period.
You put it all together.

 

[Peachyyy]

ok, that helps. thank you. I'm still kind of confused though, because I found another example that did the same thing, but based on a full house instead. They did C(4,3) C(4,2) * 13 * 12 . Any idea where the 13 * 12 came from?

 

[pka]

There are 13 ways to choose the denomination used for the three of a kind.
Then there are 12 ways to choose the denomination used for the two of a kind.
If we write the answer as 13*C(4,2)*12*C(4,2) then the process is clear.