Bridge Probability Question

[cargobeep]

On a random hand, what is the probability that you and your partner have an 8-card or greater fit in some suit?

[ArtK78]

I will leave this to others to calculate. But it seems to me that the basic problem is how many random assortments of 26 cards will have at least 8 in one suit? Obviously, the percentage will be quite high, as the only assortments of 26 cards that do not have at least 8 in one suit are 7-7-6-6 and 7-7-7-5 (in any order).

[hrothgar]

I'm lazy, so I wrote a dealer script which estimates that the probability is approximately 15.73%

 

seven =

spades(north) + spades(south) <= 7 and

hearts(north) + hearts(south) <= 7 and

diamonds(north) + diamonds(south) <= 7 and

clubs(north) + clubs(south) <= 7

 

action

 

average seven

[campboy]

Yes, about 15.736% that you don't.

 

You may as well shuffle your hand with partner's as it makes no difference who has what.

There are combin(13,7)=1716 different ways to have 7 spades. Likewise there are 1716 different ways to have 6 and 1287 ways to have 5. There are 6 different patterns for 7,7,6,6, and 4 patterns for 7,7,7,5. So the total number of ways is 6*1716^4+4*1287*1716^3. Then divide by combin(52,26) which is the number of possible sets of 26 cards the two of you can have.

 

The chance that neither pair has an 8-card fit is about 10.49%

[nige1]

# Trying to learn Dealer Script and confirming Hrothgar's result
NStrumps = 8
EWtrumps = 8
NSfit =
spades(north) + spades(south) >= NStrumps or 
hearts(north) + hearts(south) >= NStrumps or
diamonds(north) + diamonds(south) >= NStrumps or 
clubs(north) + clubs(south) >= NStrumps
EWfit =
spades(east) + spades(west) >= EWtrumps or 
hearts(east) + hearts(west) >= EWtrumps or
diamonds(east) + diamonds(west) >= EWtrumps or 
clubs(east) + clubs(west) >= EWtrumps
action average NSfit, average EWfit, average NSfit and EWfit

Yes

About 84% of deals: you have have an 8+ fit.

About 79% of deals: both sides have an 8+ fit.

Test: changing NStrumps to nine

About 38% of deals: you have have an 9+ fit.

If you have a 9+ fit, then opponents have an 8+ fit.