Math Help

[PrinceNep]

Hi All,

 

I have a quick question in regards to percentages and likelihood of breaks.

 

I'm using the website: http://www.automaton.gr/tt/en/OddsTbl.htm

 

It seems to be a very useful tool. When I was goofing around with this, however, I came upon a peculiarity.

 

When I enter Qxxx, there are 4 types of hands which have a 1 frequency.

 

I'm assuming (and this is what I'm attempting to confirm) that the site also takes into consideration that 3-1 breaks are more likely than 4-0 AND/OR the theory of empty spaces.

 

Can someone give me the mathematical backing to get to the percentage for a Qxxx opposite void (suggested probability of 4.783) verses Q opposite xxx (suggested probability of 6.217).

 

The more information/explanation you can provide the better.

 

Thank you for your help in advance!

[helene_t]

Given that xxx are in one hand, there are 23 remaining slots for the queen, 10 of which lead to Qxxx/void and 13 of which lead to Q/xxx. So the ratio between the two probabilities must be 13/10. 13/10*4.783 = 6.218 but the descrepancy is probably just a round-off error.

[ArtK78]

The website that you are using correctly calculated the possibilities of various breaks with nothing known about the opponents' hands (i.e., there are 13 empty spaces in each of the opponents' hands).

 

The odds of a 4-0 break in a given direction is 4.783%. That makes the chance that the suit will break 4-0 either way 9.566%

 

The overall odds that the suit will break 3-1 in either direction is 49.738%. However, the chance that the 1 will be the Q is 1/4 of that, or 12.434%. The chance that the singleton Q will be with one opponent is 6.217%, and the chance that it will be with the other opponent is the same 6.217%.

 

The hands with a one frequency are the two 4-0 breaks and the two 3-1 breaks with a singleton Q. Frequency in this context means how many possibilities there are of a particular distribution occuring. Given that a 3-1 break in the suit (regardless of the location of the Q) is more than 4 times as likely than a 4-0 break, the fact that the two 3-1 breaks with the singleton Q occur only in 1/4 of the possible 3-1 breaks (frequency 2 out of 8) still means that the two 3-1 breaks with the singleton Q occur more frequently than either of the two possible 4-0 breaks.

 

The break probabilities shown above are basic probabilities that most bridge players are familiar with. Roughly speaking, the chance of a 2-2 break is 40%, a 3-1 break is 50%, and a 4-0 break is 10%.

 

Where things get interesting is when you know something about the opponents' hands. That is where empty spaces comes into play. The website you are using allows for you to adjust for empty spaces.

 

Suppose LHO opens 2♠ at both vul and you wind up in 4♥ in a 9 card fit missing the Q. Suppose your combined spade holding is 5 cards. You can assume with reasonable certainly that the 2♠ bidder has 6 spades and his partner has 2 spades. Adjusting for empty spaces (7 in the 2♠ bidder's hand and 11 in his partner's hand), the odds of the various breaks in the heart suit change. Now the chances of a 2-2 break are 37.736%, the chances of a 3-1 break are 50.326% (3 with the spade bidder 12.581%, 1 with the spade bidder 37.745%) and the chances of a 4-0 break 11.928% (4 with the spade bidder 1.144%, 0 with the spade bidder 10.784%). The odds that the opponent short in spades has the heart Q is 61.106% (Qxxx - 10.784%, plus 3/4 of the 3-1 breaks where he holds the three card holding, or 3/4 of 37.745% = 28.309%, plus 1/4 of the 3-1 breaks where he holds a singleton, or 1/4 of 12.581% = 3.145%, plus half of the 2-2 breaks, or 1/2 of 37.736% = 18.868%).

[MFA]

We can calculate the probabilities like this:

 

Let's call the suit Q432.

 

The probability that west has the Q is 13/26 since west and east each have 13 empty slots.

 

Given than west has the Q, then the probability that west also has the 4 is 12/25, since west now has 12 empty slots and east 13 empty slots.

 

Etc., and we get

Probability that west has Q432 is 13/26 * 12/25 * 11/24 * 10/23 = 4,783%

 

Analyzing Q to 432 is similar:

 

West has the Q: 13/26

East has the 4 (given west has the Q): 13/25

East has the 3 (given west has the Q and east the 4): 12/24

East has the 2 (given west has the Q and east 4-3): 11/23

 

Probability that the suit is Q to 432 is 13/26 * 13/25 * 12/24 * 11/23 = 6,217%