[billw55]

The thread about the probability of a 3-2 break prompted me to post this.

For years I have been bothered by the 68% figure. This is a problem where is it quite easy to list all the outcomes. There are 16 ways to divide 5 cards into two hands. 10 of these ways are 3-2 divisions. 10/16 = 62.5%.

And yet in books, web sources, forum posts, etc, the probability of a 3-2 break, in isolation, is always given as 68%. What gives? I must be missing something obvious …

[hrothgar]

delete

[inquiry]

billw55, on 2011-September-23, 13:41, said:

The thread about the probability of a 3-2 break prompted me to post this.

For years I have been bothered by the 68% figure. This is a problem where is it quite easy to list all the outcomes. There are 16 ways to divide 5 cards into two hands. 10 of these ways are 3-2 divisions. 10/16 = 62.5%.

And yet in books, web sources, forum posts, etc, the probability of a 3-2 break, in isolation, is always given as 68%. What gives? I must be missing something obvious …

You get your 62,5% by only considering the combination of the five cards
5,0 = 1 way
5.1 = 5 ways
5.2 = 10 ways
5.3 = 10 ways
5.4 = 5 ways
5,5 = 1 way

This is 32 combinations of which 20 are 3=2, The problem is you do not consider the other 21 cards the other side holds.

When one opponent holds a void, there are 203,490 unique hands they can hold. When they hold a singleton in the suit, there are 1,469,650 different hands they can hold, when they hold 2 cards, 3.537,160. these sequence then repeats for when they hold 5, 4, and 3 cards in the suit.

Thus instead of dividing 20 by 32, you have to divide (3,537,160 x 2) by 10,400,600 which comes out to be essentially 68%

The calculations are:
C(21,8)*C(5,0) = 203490 x 1
C(21.9)*C(5,1) = 293930 x 5
C(21,10)*C(5.2) = 352716 x 10
C(21,11)*C(5.3) = 352716 x 10
C(21.12)*C(5.4) = 293930 x 5
C(21,13)*C(5.0) = 203490 x 1

Hope that helps.

[aguahombre]

Someone, about a year ago explained how this this works, but I forgot. Maybe it was Inquiry, then also.

Same thing happens with the 4-3 split missing seven in a suit, which to us simple souls seemed about 52%, but is more.

[billw55]

Ah ok right Ben

There are more ways to divide the *other* cards when the suit breaks evenly.

Got it now

[manudude03]

the way I do it is 2 x 5C3 x 13/26 x 12/25 x 11/24 x 13/23 x 12/22 which comes to 0.67826..... rounded off to 68%.