A little bit of math

[Poky]

1♥ 2♥ pa 2♠

pa pa ??

 

We have 3163.

 

If we fix:

Partner: 5+♥

RHO: 5+♠5+m

LHO: 2-3♠

 

What is the percentage that RHO holds clubs, as his minor suit?

By intuition, I would say ~85%, but what says the math?

[rogerclee]

I don't know about the actual numbers, but we can do a quick calculation as follows. Assume that partner has 5 hearts, LHO has 2 spades, and RHO has 5 spades.

 

This means that there are 3 spades, 7 hearts, 7 diamonds, and 10 clubs we don't know about (27 cards). This also means LHO has 11 cards, partner has 8 cards, and RHO has 8 cards we don't know about.

 

We deal RHO 8 cards from the 27 card "deck". We can count the number of ways to deal him a certain number of a minor. An example calculation for "probability he has 5 clubs" is

 

binomial(10,5) * binomial(17, 3) / binomial(27,8)

 

This is the number of ways to choose 5 clubs, multiplied by the number of ways to choose his remaining 3 cards, divided by the total number of possible 8-card hands.

 

CLUBS

5: .077

6: .0129

7: .0009

8: 0

TOTAL: .0908

 

DIAMONDS

5: .0108

6: .000599

7: 0

TOTAL: .01679

 

Given that one of these events occurred then, it is 84.4% that he had clubs rather than diamonds.

 

The same calculation by assuming RHO has 6 spades (LHO has 2 spades, RHO has 5 hearts still) yields a 90.1% chance that he has clubs.

 

What is the actual number? In general having more known cards (like knowing LHO has 3 spades or partner has 6 hearts), as well as increasing RHO's number of spades, will increase the probability that he has clubs. The real number is probably hovering around 87-88.

[helene_t]

Very nice, Roger.