Bridge-Math Sanity check

[OleBerg]

You don't even get to see any hands. The opponents bid:

 

1♣ - 1♦

4♥

 

1♣ = Two-way; either 15-19 bal., or natural. (The balanced type cannot contain 5-card major.)

1♦ = Transfer-Walsh, shows 4+ hearts.

4♥ = Four-card support and either a balanced hand or specifically 2-4-2-5. In both cases "18-19".

 

Now, my claim is that the a priori probabilety of the 4♥ bidder holding a balanced hand is 87,5%.

 

The 4♥ bidder can have these shapes:

 

4-4-3-2

4-4-2-3

4-3-4-2

4-3-2-4

4-3-3-3

4-2-4-3

4-2-3-4

2-4-2-5

 

So one shape in eight is the non-balanced type.

 

 

Just to specify how I (a non-mathematician) uses the word a priori here:

 

It means before adjustments for:

 

- There are more "third spaces" in the balanced hand, making it ever so slightly more easier to get a high point count. (Adjusting the number slightly opwards.)

- It is more likely that a 2-4-2-5 hand will be considered "18-19". (Adjusting the number slightly downwards.)

- Inferences from the bidding, like the fact that responders hearts are as long, or longer, than the spades. (I have no idée if this would adjust up or down.)

 

I know these adjustments would all be very small, they are just mentioned for the sake of completeness.

[cherdano]

2=4=2=5 is less likely than any of the other specific shapes.

[zasanya]

I wrongly voted u r rt.The actually probability of precisely 2-4-2-5 seems to be 6.3% and the combined probability of other 7 events 93.7% .This is so because as cherdano points out the probabilities of all 8 events is is not same.If you make 8 slips and write out these combinations and pick up any one slip then your number would be rt because the probability of any one slip being picked up is same.But here all the events do not have same probability of occuring.

[Fluffy]

2425 is infinitelly more likelly than 4234

 

EDIT: I think I understand now that the suits ain't on the classic order

[OleBerg]

2425 is infinitelly more likelly than 4234

 

EDIT: I think I understand now that the suits ain't on the classic order

 

 

Yes, somehow I put ♥ before ♠. Let's say it was because they were thrumphs. :rolleyes:

[han]

Or trumps.

[TWO4BRIDGE]

Yes, somehow I put ♥ before ♠. Let's say it was because they were thrumphs. :rolleyes:

Is the 2 4 2 5 shape ( your last example ) really 2h, 4s, 2d, 5c ?? or was it supposed to be 4 2 2 5 ( 4h, 2s, 2d, 5c ) in your bastardized nomenclature ?

[kayin801]

I think you mentioned this, but opener is much more likely to stretch with a good 4225, espcially with a good 5 card club suit, maybe with only 16 pts like Ax, A10xx, xx, AKJ10x, so I think the range of hands that can be in that shape is wider. Agree with what others said though about 4225 being less likely as a shape.

 

I don't think you can make any bidding inferences practically here about responder's hand.

[ceeb]

The literal answer to your question is that you forgot one big thing -- the unequal probabilities of the various distributions. For 2425 I make the chance to be 0.0617 or nearly half of 1/8. In stipulating to ignore the extra comparative chance that 2425 would be worth bidding 4♥, you stipulated away a compensating advantage that could be quite large.

[Fluffy]

chances to have 18 HCP with a 4333 are larger than with 5422

[hotShot]

You don't even get to see any hands. The opponents bid:

 

1♣ - 1♦

4♥

 

1♣ = Two-way; either 15-19 bal., or natural. (The balanced type cannot contain 5-card major.)

1♦ = Transfer-Walsh, shows 4+ hearts.

4♥ = Four-card support and either a balanced hand or specifically 2-4-2-5. In both cases "18-19".

 

 

The numbers indicate that you play a 12-14 NT.

Does that mean that you open balanced 12-14 hands with 3 cards in ♣ with 1NT and that you weak NT never has a 4/5 card club suit?

[gnasher]

I agree with ceeb

 

4432 types: 0.2155 * 6/12

3433: 0.1054 * 1/4

2425: 0.1058 * 1/12

 

(0.1058 * 1/12) / (0.2155 * 6/12 + 0.1054 * 1/4 + 0.1058 * 1/12) = 0.062

[ceeb]

chances to have 18 HCP with a 4333 are larger than with 5422

That's true. The chance of >17hcp is about double with the balanced distribution. But as against that it looks to me that 18-19hcp 3433 facing a minimum 1♥ response tends to be a poor game.

 

If you consider instead the criterion that opener have a 5-loser hand, then the situation is more than reversed: In my simulation of 200000 deals (opener 5 losers and one of the listed distributions, responder 4+ hearts and either shorter spades or at most 4 spades), four times as many 2425 hands qualified (38 hands, 22% of the total, nearly 4-fold over-representing expectation based on distribution alone) as 3433 hands (5% of the total, more than 3-fold under-representing the 18% expectation based on distribution alone -- and all 9 of these hands have >19hcp, so nominally don't even open 1♣).

 

I suppose the true rate of 4♥ bids is somewhere between the extremes of a rabid point-counter and a LTC zealot, but still expect that the expectation for 2425 is well ABOVE 1/8.

[OleBerg]

The numbers indicate that you play a 12-14 NT.

Does that mean that you open balanced 12-14 hands with 3 cards in ♣ with 1NT and that you weak NT never has a 4/5 card club suit?

 

No. Any balanced 12-14 open 1NT. (Unless it has a 5-card major, which is opened one of the suit.)

[whereagles]

Echo the above comments. The various shapes don't have the same probability.

 

Didn't reproduce zasanya's calc, but it sounds just about right.