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Card Hands

Date: 05/26/97 at 23:02:17
From: Huyen Nguyen
Subject: Probability

My teacher assigned me a probability problem that I just don't know 
how to solve. I need to find the probability of getting a straight 
(5 cards in a row) in a deck of cards. I also need to find the 
probability of getting a royal straight, where only possible cards are 
10, jack, queen, king and ace. Finally, I need to find the probability 
of getting a royal flush (cards with the same suits).  

I have no clue to how I would find this. I even went out to buy a pack 
of cards to try it myself.  But that doesn't seem to work at all!  
Please help.  

Thank you so very much :-)

Date: 05/27/97 at 05:33:06
From: Doctor Mitteldorf
Subject: Re: Probability

Dear Huyen,
Getting the cards is a good start, except that straights happen so 
rarely that you might be dealing to yourself for a long time before 
you got one, and in order to estimate the probability you need 3 or 4.  
A computer program that can do something analogous to shuffling and 
dealing a deck of cards 1,000,000 times in a few seconds would be 
better. This is actually a legitimate, established method from 
probability science, and it's used for hard problems that you can't do 
accurately any other way.
Your problem, however isn't that hard. I don't know what you've 
learned about probability in class already that might get you started, 
but here's a way to think about it.
First, suppose that you had to get all 5 cards for the straight from 
the shuffled deck IN ORDER.  The first card could be anything from 2 
to 10, because if you start higher than 10 you don't have room to make 
a straight. There are 36 cards out of the 52 that would be okay for 
your first, and that means a probability of 36/52 for your first step.  

Having reached that step, your next step is more determined: the next 
card has to be just one higher. There are 51 left in the pack, but 
only 4 will do. So far, the probability of getting the first two cards 
right is 36/52 * 4/51.  

For the next step, there are four good cards out of a pack of 50, so 
the probability of getting the first three good is:

36/52 * 4/51 * 4/50  

Keep going this way on your own and finish to the fifth card.
But now you're not quite done because you asked the question about 
getting all 5 cards for the straight IN ORDER and the question your 
teacher asked was about the probability of getting the same 5 cards IN 
ANY ORDER. The real answer is x times bigger, where x is the number 
of possible orders that you can put 5 cards in. How many different 
orders can you put 5 cards in?  

Orders in probability theory are called "permutations". You may have 
learned that there are 5*4*3*2*1 different ways of ordering 5 cards, 
and that this number is called 5! or "5 factorial".  If you haven't 
already learned this, see if you can figure out why it would be so.
Now you're ready to write down the answer for a straight, and multiply 
out all those numbers to see what you get. Please write back and tell 
me your answer. Also, let me know how you do with those other 
questions about the royal straight, etc.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!   
Associated Topics:
High School Probability

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