Wheel in Poker

[Whatsmytotal]

I'm sure this is commonly known but I'm having trouble getting the answer...

What are the chances of drawing a wheel, A2345 (in any order), in 7 cards from a standard deck?

[moo321]

Ugh. Kind of long to calculate. Odds of catching it in 5 cards + all different varieties of draws to it, etc. Why?

[bj bob]

Originally Posted by moo321

Ugh. Kind of long to calculate. Odds of catching it in 5 cards + all different varieties of draws to it, etc. Why?

He's probably talking about 7 stud lowball, so all he needs to know is just that, the probability of 5 wheel cards drawn with a 7 card hand.

[moo321]

Originally Posted by bj bob

He's probably talking about 7 stud lowball, so all he needs to know is just that, the probability of 5 wheel cards drawn with a 7 card hand.

Well, it's just that it's kind of hard to calculate. You have to calculate the odds of making it in 5 cards, then the odds of each possible hand with any draw to it after 4 cards (i.e. contains 2-4 low cards), then the odds that those draws will pan out. Very time consuming, unless you can somehow make a computer program to do it...

Only other possibility is that I remember there being some lowball stats in Supersystem. Maybe there's something in there?

[miplet]

According to wikipedia, it is 781,824 out of 133,784,824 (or about 1 in 170) hands. I tried doing the math myself but gave up after 20 minutes. I'll try again after I post this.

[RG1]

According to my quick calculations, about 15 minutes.

You'll get a wheel in the first five cards once every 2538 hands.
You'll get a wheel in the first six cards, including the first five, once every 468 hands.
You'll get a wheel in the first seven cards, including the first five or six, once every 113 hands.



What I did was take 20/52 * 16/51 * 12/50 * 8/49 * 4/48 * 47/47 * 46/46 = 64/162435 = 1/2538.047 = prob of a wheel in the first five cards.

Then for the first six cards I did 20/52 * 16/51 * 12/50 * 8/49 * 48/48 * 4/47 * 46/46 = 1024/2544815. Then put the nothing card in each of the other six spots, added up the totals for 11223/5253518 = 1/468.1.

For seven cards I did each of the 21 possibilities to make a wheel out of seven cards, added them up for 58581/6632282 = 1/113.26.


I'm concerned I may have made a clerical error somewhere since it doesn't add up to the 1/170 the (edit - wikipedia not wizard) got, so somebody double check.

[RG1]

I found my mistake from earlier. The cards that don't make a wheel aren't 47/47 or 46/46, they should be 43/47 or 42/46 or whatever depending on where they are.

Therefore,

First five cards = 1/2538
First six cards = 1/503
Seven cards = 1/171

[moo321]

This sounds like a good estimate, but I don't think it's totally accurate.

[bj bob]

Originally Posted by moo321

This sounds like a good estimate, but I don't think it's totally accurate.

Well then Moo, you know the next step. Happy calculating!