[WellSpyder]

Any statisticians out there who know the standard deviation of the number of (Milton Work) points held in a bridge hand? I played a 24 board match last week in which my partner averaged 8.25 points and I averaged 8.125. My partner commented that our combined average of 16.375 was "statistically off the scale", and I just wondered how far off the scale it really was...

Incidentally, we also held a grand total of zero 7 card suits or longer - not surprisingly, perhaps, we seemed to end up doing an awful lot of defending....

[PeterAlan]

For a 13-card hand in isolation it's 4.130. I don't know the number for a pair of hands within a deal.

[barmar]

If the hands were independent, it would be relatively straightforward to compute the combined SD (if I understand the wikipedia article, it's the root mean square). But when one player is weak, it presumably means the other players, including his partner, are likely to be stronger, so they're not independent.

[weejonnie]

This is a guess: If for 13 cards it is 4.13 then for 26 cards it would be 2.92.
But for 24 boards it would be 0.596. (Mean of 20 of course)

So a total of 16.375 is 6.08 SDs away from the mean - not sure what the value is - something like 1 in a million or 1 in 10 million probably.

If this sounds a lot - remember that there are probably several thousand (if not tens of thousands) of bridge events daily worldwide - If we assume 10,000 then such an eventuality would occur once every 1000 weeks (20 years) at the lower end or once every two years at the higher.

[WellSpyder]

Thanks, everyone. Some useful pointers, there.

I thought an approach like weejonnie's would understate the odds against what happened, since, as barmar points out, the HCP average of my partner's and my hands aren't independent - if partner averages 8.5, for instance, I can expect to average 10.5 rather than 10. However, it seemed a good place to start. The problem is that, according to Excel at least, the chances of getting an outcome from a Normal Distribution as far away from the mean as 6.08 SDs is about 6E-10. In other words, 1 in about 1.7 billion! I guess my assumption that it would be OK to use a normal approximation for what obviously isn't exactly a normal distribution breaks down at such extreme values....

Anyway, I stand by the title of this thread!

[PhilKing]

I think I've been running at about 13 for the last few months at rubber bridge, but I have a sneaking suspicion it won't last.

In fact, I have probably just bocked myself.

[helene_t]

The SD on a the HCPs in two hands on a single board is 4.77. For the average across 24 boards it is 0.97. So you were off by almost four SDs. That is very unlucky (or maybe lucky, if your strongest skill is defense ).

I calculated this by tabulating all the honour combinations but now that we are told that the SD of the HCPs of a single hand is 4.13 it could be done simpler. We know that the correlation coefficient between EW and NS is -1 and also that the correlation coefficient between say ESW and N is -1. From additivity of covariances is follows that the variance of the HCPs of a single hand is -3C and the variance of the sum of the HCPs of two hands is -4C, where C is the covaraiance between the HCPs of two hands. So the ratio between the two-hand SD and one-hand SD is sqrt(4/3)

[jogs]

helene_t, on 2015-June-23, 06:06, said:

The SD on a the HCPs in two hands on a single board is 4.77. For the average across 24 boards it is 0.97. So you were off by almost four SDs. That is very unlucky (or maybe lucky, if your strongest skill is defense ).

There was the added condition that both partner's averaged under 8.5 points. Does that make it closer to 3 Std dev.?

[WellSpyder]

Brilliant! - thanks, Helene. That sounds pretty convincing to me. 3.74 SDs below the mean occurs around 1 time in 11000 in a Normal Distribution, so with luck if I play once a week it won't happen again for another 200 years or so....

(No 7 card or longer suit for either of us over 24 boards is likely to happen around 1 time in 7, I think. Putting the two probabilities together - I know they won't be strictly independent, but not far off, I would guess - suggests I should be OK for the next 1400 years or so.)

[WellSpyder]

PhilKing, on 2015-June-23, 05:30, said:

I think I've been running at about 13 for the last few months at rubber bridge...

When I told someone last night that I was trying to calculate the odds of averaging only 16.375 points between my hand and partner's over 24 boards, their immediate response was that it depended on whether I was playing rubber bridge or not.

Sounds like it might also depend on who I am playing against...

[PeterAlan]

This got me interested enough to generate the actual HCP distribution for a pair of hands, and work directly from those numbers. Initially I thought to myself that, since the HCP distribution of a single hand is obviously not symmetric about the mean, then the HCP distribution of two also might not be, but then I remembered that for every pair of hands with N points there's another with (40-N)! So the distribution is indeed essentially Normal, with mean 20 and SD, as Helene has already given much more elegantly, of 4.77 [4.769182].

Since I had the numbers, I let the machine compound them up for 24 deals, and the (E&OE) number-crunched result I got for WellSpyder's combined holding of a total of 393 (or less) HCPs between two hands over 24 boards was 0.010260% (1 in 9747).

[barmar]

So if you play an average of 1 session of bridge/day, it's about once every 27 years. So it's about twice in a lifetime.

If both of those happen to be with the same partner, the bridge gods really have it out for you two. You probably shouldn't travel together to any tournaments.

[WellSpyder]

PeterAlan, on 2015-June-24, 09:51, said:

This got me interested enough to generate the actual HCP distribution for a pair of hands, and work directly from those numbers. Initially I thought to myself that, since the HCP distribution of a single hand is obviously not symmetric about the mean, then the HCP distribution of two also might not be, but then I remembered that for every pair of hands with N points there's another with (40-N)! So the distribution is indeed essentially Normal, with mean 20 and SD, as Helene has already given much more elegantly, of 4.77 [4.769182].

Peter, I've got another probability question which your HCP distribution for a pair of hands should be able to answer. What is the probability of a combined pair of hands having 25 HCP or more? (Is the Normal approximation accurate for this question?) Working from this, I should be able to work out the probability of playing a 32-board match in which my partner and I never held as many as 25 HCPs between us, as we did last Sunday....

[Zelandakh]

WellSpyder, on 2016-April-21, 04:04, said:

Peter, I've got another probability question which your HCP distribution for a pair of hands should be able to answer. What is the probability of a combined pair of hands having 25 HCP or more? (Is the Normal approximation accurate for this question?) Working from this, I should be able to work out the probability of playing a 32-board match in which my partner and I never held as many as 25 HCPs between us, as we did last Sunday....

This information is available online, for example here. The answer given for 25+ is around 17.5% (I have not personally checked this figure).

[helene_t]

Zelandakh, on 2016-April-21, 04:43, said:

This information is available online, for example here. The answer given for 25+ is around 17.5% (I have not personally checked this figure).

The normal distribution approximation (based on 24.5+ being rounded off to 25+) is 17.3% so that is not too far off.

The HCP distribution has smaller kurtosis than the normal approximation, i.e. although the variance is (by definition) the same, the chance of both you and p getting a yarb is one out of two millions, while the normal distribution approximation says it should happen once every 76,000 boards. The flip side of this is that the normal distribution also overestimates the chance of you and p getting exactly 20 HCPs together: 8.35% vs 8.22%.

[WellSpyder]

Thank you both! So this will happen about once in every 470 matches, I think. Disappointing, but not that exceptional.

Not surprisingly, we didn't bid that many games - and only made one of the three that we did bid.....

[PeterAlan]

WellSpyder, on 2016-April-21, 04:04, said:

Peter, I've got another probability question which your HCP distribution for a pair of hands should be able to answer. What is the probability of a combined pair of hands having 25 HCP or more? (Is the Normal approximation accurate for this question?) Working from this, I should be able to work out the probability of playing a 32-board match in which my partner and I never held as many as 25 HCPs between us, as we did last Sunday....

I've just seen this, and the numbers in the link Zelandakh quoted are the same as I have (as one would hope!). So it's (1) 17.5375% (to 4 dp), and (2) once in every 478+ 32-board matches.

PS: Since the probability of one side holding 16 to 24 points in a deal is 64.9249%, the probability of neither side holding 25+ points for an entire 32-board match is just under 1 in 1 million.

[billw55]

So how did your teammates fare while averaging 24 points per board?

[WellSpyder]

billw55, on 2016-April-21, 07:08, said:

So how did your teammates fare while averaging 24 points per board?

Teammates didn't average that many points, only around 21.3 per board. But they did have 7 hands with 25+ points between them. And they did very well, with two making slams, 8 making games and just two failing games.

[manudude03]

I once had a 28 board match where partner and I averaged 17 between us. We bid 1 slam which made and 15 games of which 9 made, funny game