Two Aspects of Draw: Ben Frisch

[Al Mirpuri]

There is an article called 'Two Aspects Of Draw' by Ben Frisch in the Internet Magazine. He states that it is 44% to improve a pair with a three card draw to two pairs or better. Mike Caro gives a comparable table at www.poker1.com that gives it at being 28.71% to make two pairs or better (I say comparable as Frisch does not specify what his pair is whilst Caro has it as being aces). Who is right? Who is wrong? My experience tells me that Frisch is wrong.

Anyone say differently? Please post.

[MarkGritter]

Here's the quote:

[ QUOTE ]

The chances of making two pair or better if drawing three cards to a pair is about 44% or about (1-(39/47 * 38/46 * 37/45) = 1-(.8298 * .8261 * .8222) = 1-.5636 =43.64%) or odds of about 9-4. So by keeping that ace you are significantly hurting your chances to improve your hands.


[/ QUOTE ]

The calculation appears to be "how often will 8 outs hit" which doesn't make much sense to me.

I think if you dig through the archives you can find examples calculating the number exactly; Caro's number is correct.

Quick and dirty estimate #1: use twodimes.net calculator in 2-7 lowball mode. AAKQJ pat loses to 88/743 drawing three 29.7% of the time in lowball so the pair of eights wins 29.7% of the time playing for high. This is in line with the Caro number (not exact because we specified 5 more cards), obviously not anywhere near 44%.

http://twodimes.net/h/?z=2941094

Quick and dirty estimate #2: 47 unknown cards

On the first card you have 2 outs to improve.
On the second card you have 4 or 5 outs to improve.
On the third card you have 6, 7, or 8 outs to improve.

Assuming the best case you will brick 45/47 on the first card, 41/46 on the second card, and 37/45 on the third card. So your chances of improving must be less than 1-45/47*41/46*37/45 = 29.83%.

The worst case is bricking 45/47 on the first drawn card, 42/46 on the second card, and 39/45 on the third card, or no less than 1-45/47*42/46*39/45 = 24.34% to improve.

[2461Badugi]

[ QUOTE ]
Here's the quote:

[ QUOTE ]

The chances of making two pair or better if drawing three cards to a pair is about 44% or about (1-(39/47 * 38/46 * 37/45) = 1-(.8298 * .8261 * .8222) = 1-.5636 =43.64%) or odds of about 9-4. So by keeping that ace you are significantly hurting your chances to improve your hands.


[/ QUOTE ]

I think if you dig through the archives you can find examples calculating the number exactly; Caro's number is correct.


[/ QUOTE ]

Interesting bit bolded. 44% sure ain't 9:4, but Caro's number is pretty close to it. Someone read odds notation backwards and tried to self-justify?

[MarkGritter]

Caro's number is closer to 10:4, though:

71.29 : 28.71 = 2.48 to 1

[VickreyAuction]

That caught my eye too.