Expert: Sherry Wallin - 7/28/2011

Question
hello! just wanna ask some questions, that my head really hurts everytime i try to compute the probabilities..i am a student of a university here in the Philippines and my professor assinged me to report about probability and permutations of the ff:
1) Of getting a pair of cards in a standard deck of cards?
2) For 3-of-a-kind?
3) For 4-of-a-kind?
4) For Royal Flush?

hope for your immediate response.. thanks!
and more power :)

Answer
Sanny~

What is ff?

1 - the probability of getting a pair of cards in a standard deck of cards. Well how many pairs are there in a standard deck? To figure this out you will need to take each number and figure out how many pairs can be made. For instance there are 4 twos, 2hearts (2h), 2spades (2s), 2diamonds (2d), and 2clubs(2c)
Once you choose the first card there are 3 ways to make a pair, so let's say you choose the 2h, then in order to make a pair you would have 3 cards (ways) to pair up: 2h,2c; 2h,2s; 2h,2d and if you chose the 2c then you would have 2c,2h; 2c,2d; 2c,2d (noting that 2h,2c and 2c,2h are the same), and if you chose the 2d, then you could make 2d,2c; 2d,2h; 2d,2s; noting again that 2d,2c and 2c,2d are the same pair AND 2h,2d and 2d,2h are the same also. And finally your last first choice would be 2s,2h; 2s,2d; 2s,2c noting that all are repeats, so you really have 6 pairs from 4 like numbered cards. Now take into account that there are 13 sets of 4 of a kind in a standard deck of cards so there are 13*6 = 78 pairs in a standard deck of cards.In order to find the probability you need to figure out how sets of 2 there are in a standard deck of cards and then take 78 and divide it by that number. Now this where I will use combinations to show you.

What I did the long way above is I did 4C2, i.e., four things taken two at a time or 4!/2!2! or 4*3*2!/2!2! = 6 then using either permutations or the multiplication rule I did 13 sets times 6 = 78
To find the number of 52 things taken 2 at a time = 52C2 = 52!/2!50! = 52*51*50!/2!*50! =52*51/2 =26*51 = 1326 sets of two cards in a standard deck of playing cards thus 78/1326 will be the probability of getting a pair in a standard deck of playing cards: 78/1326 ~ .0588

Math Prof