Probability: 4 of a kind

[Prashant Ranjan]

In the poker game Five-Card Draw, each player is dealt a hand consisting of 5 cards from a deck of 52 cards. Each card in the deck has a suit (clubs, hearts, diamonds, or spades) and a value (A, 2,..., 10,J,Q,K).
What is the probability that we are dealt with four of a kind?

An expert help would be appreciated.

Thanks
Prashant

[Ian Stewart]

You will *never* be asked this kind of 'poker question' on the GMAT, though they are very popular in undergraduate combinatorics classes. For one thing, these questions are generally quite a bit too complicated for the GMAT, and for another, they rely too heavily on knowledge that not all test takers will share (the composition of a deck of cards). I'll give a solution, but you really don't need to worry about it for the test. If you aren't at all familiar with cards, that's even more reason to skip this question.

In total, there are 52C5 ways to choose a set of 5 cards out of 52. That's the denominator of our probability (where 52C5 = 52*51*50*49*48/5! ).

In the numerator, there are 13 different ways to choose the set of cards which will be our "4 of a kind" (they can be all of the aces, or all of the kings, etc). Once we choose our 4 of a kind, there are 48 cards left from which we can choose our last card. So there are 13*48 hands you can make which contain "4 of a kind". That's the numerator of our probability.

So the answer is 13*48/52C5, which, if you simplify, turns out to be roughly 1/4000.

[Jim@StratusPrep]

To find the number of ways you can select 5 cards it is 52 combination 5 or 52!/(5!)(47!) = 2,598,960

There are then 13 different cards you can have 4 of a kind of and 48 other cards to go with it or 13 x 48 = 624

624/2,598,960 = 1/4165

There are some simpler ways to do the math, but this should give you an idea. Unlikely you will see this on the GMAT with the size of the numbers.

Easier math

(13 x 48)/(52 x 51 x 50 x 49 x 48)/(5!)

simplifies to 1/(4 x 51 x 50 x 1)/5!... just looking at the denominator now: (4 x 51 x 50 x 49)/ (5 x 4 x 3 x 2) simplifies to 17 x 5 x 49.

GOOD LUCK!!!

[Prashant Ranjan]

Thanks Ian and Jim. Got a great relief on knowing that cards questions don't come in GMAT. I also think that a person unfamiliar with the cards would be perplexed in these sort of questions.

Thank you guys.
Regards
Prashant