5 draw stripped deck advice needed - Page 2

archmagi · #11 12-02-2003, 07:45 AM

Hi Buzz,
I'm with you on the trips. It's the calculation of the straight that I'm looking into.
Give me a couple of days on this one, and I'll post my calculation.
Again, thanks for the input

Buzz · #12 12-02-2003, 10:20 AM

Hi Heretic - If you’re with me on trips, then straights should be easy to explain.

I do agree that you should spend some time and work it out for yourself.

But if you get stuck, here’s a way to think:

Let’s momentarily forget about suits. We can subtract for straight flushes later. And if we forget about suits, then we have only four possibilities for straights:

AKQJT
KQJT9
QJT98
JT987

Right? (If you can think of another one, what is it?)

Let’s think about picking cards out of a deck to make one of these, but the reasoning will be the same for all of them. Let’s think about picking cards out of a deck to make KQJT9.

What cards do we need to pick out of the deck (either a shortened 32-card deck or a 52-card deck) to make KQJT9? Well... KQJT9 needs a king, a queen, a jack, a ten, and a nine. That’s all it needs - and that’s exactly what it needs. Nothing more, nothing less.

Now how many kings are there in a deck? However many there are, that’s the number of ways you can take a king from a deck and put it in the hand you’re building. Similarly, how many queens are there in a deck? How many jacks? How many tens? How many nines?

There are four kings, four queens, four jacks, four tens, and four nines.

But any one of the kings can go with any one of the queens. For example, the king of spades can go with the queen of spades, the queen of hearts, the queen of diamonds, or the queen of clubs. Thus there are four ways you can put a queen with the king of spades.

Then there are four ways you can put a queen with the king of hearts. And so forth. You should almost immediately see that the number of ways you can put a king and a queen together is 4*4.

Choose any one of those king-queen combinations, the king of clubs and the queen of hearts, for example. There are four ways you can put a jack with that combination. That makes 4*4*4.

And if you take any particular one of those three-card combinations, for example the king of clubs, the queen of hearts, and the jack of hearts, there are four ways you can put a ten with that combination. That makes 4*4*4*4.

Finally, if you take any particular one of those four-card combinations, the king of clubs, the queen of hearts, the jack of hearts, and the ten of spades, for example, then there are four ways you can put a nine with that combination. That makes 4*4*4*4*4.

Therefore, there are 4*4*4*4*4 ways to build a KQJT9 straight from a deck of cards.

However we built a KQJT9 straight, there are exactly the same number of ways to build the other straights. Thus:

(4*4*4*4*4) for AKQJT,
(4*4*4*4*4) for KQJT9,
(4*4*4*4*4) for QJT98, and
(4*4*4*4*4) for JT987,

You can see we’re taking (4*4*4*4*4) four times, or (4*4*4*4*4)*4. That’s the total of number of different ways you can build a straight, picking one card out of a deck at a time, if you disregard straight flushes.

But since we’re listing the straight flushes in a separate category, we have to subtract the total number of possible straight flushes from the number of straights. For clubs four of straight flushes are possible: AcKcQcJcTc, KcQcJcTc9c, QcJcTc9c8c, and JcTc9c8c7c. Similarly there are four straight flushes for spades, hearts and diamonds, a total of 16 straight flushes in all.

Bottom line: 4*4*4*4*4*4-16 = the number of straights.

Hope that makes sense to you. It’s slightly different reasoning than I used in my previous post - and hopefully explains straights better. Of course the bottom line total, 4080, is the same.

I only have one book by Bob Ciaffone in my poker library. I didn’t see any glaring mistakes in it. Are you sure you didn’t misquote him?

Buzz