Needing some probabilities for playing bridge better.

Question

First of all, I would like to underline that I am very bad in probabilities and that I am a fanatic bridge player.

Yesterday, after a tournement, we had a no-end discussion my partner and I; so the question.

Let me note $n_S,n_H,n_D,n_C$ the number of spades, hearts, diamonds and clubs in a hand.

Suppose that my hand is $n_S=4$, $n_H=4$, $n_D=3$, $n_C=2$.

What are the probabilities that my partner has in his/her hand

• $0<n_S \leq 3$
• $0<n_H \leq 3$
• $n_C \geq 6$
• $n_D < n_C$
• $n_D=13-(n_S+n_H+n_C) \qquad$ (by definition)

The probality of each possible case is important for me to know.

I hope and wish that this question will not be closed because missing context.

Answer 1

Of the 39 remaining cards, there are $39!$ possible hands your partner may have, each having equal probability.

Given a specific shape for your partner, we can also count the number of hands of that shape:

$$ \binom{11}{n_\clubsuit} \cdot \binom{10}{n_\diamondsuit} \cdot \binom{9}{n_\heartsuit} \cdot \binom{9}{n_\spadesuit}$$

because the binomial coefficient

$$\binom{11}{n_\clubsuit} = \frac{11!}{n_\clubsuit! (11 - n_\clubsuit)!}$$

counts the number of ways to choose $n_\clubsuit$ cards from the 11 remaining clubs.

There are enough cases that the simplest approach is to use a computer to simply exhaust over all possible shapes your partner may have.

---

Of all possible hands your partner may have, roughly 6.1% will meet your constraints. (that number jumps to 6.6% if you made a typo and your partner is allowed to have voids in the major suits)

Among those hands, the most common are:

• 14.5% of them are 2-2-3-6
• 12.7% of them are 2-3-2-6
• 12.7% of them are 3-2-2-6
• 8.5% of them are 1-3-3-6
• 8.5% of them are 3-1-3-6
• 6.6% of them are 3-3-1-6

Of particular note, roughly 44% of the hands of the shape you describe will be semibalanced.

I've posted below the total list of data I've computed. The format is:

<odds of this shape> (<absolute odds this shape>, <clubs>, <diamonds>, <hearts>, <spades>)

where the two different kinds of odds are the proportion assuming your partner's hand satisfies your constraints, and the proportion among all hands.

1.638531801225993e-07 (9.972390705024487e-09, 11, 0, 1, 1)
7.20953992539437e-06 (4.3878519102107746e-07, 10, 0, 1, 2)
7.20953992539437e-06 (4.3878519102107746e-07, 10, 0, 2, 1)
1.802384981348593e-05 (1.0969629775526938e-06, 10, 1, 1, 1)
8.4111299129601e-05 (5.119160561912571e-06, 9, 0, 1, 3)
8.4111299129601e-05 (5.119160561912571e-06, 9, 0, 3, 1)
0.00014419079850788743 (8.77570382042155e-06, 9, 0, 2, 2)
0.00036047699626971856 (2.1939259551053875e-05, 9, 1, 1, 2)
0.00036047699626971856 (2.1939259551053875e-05, 9, 1, 2, 1)
0.0004055366208034333 (2.4681666994935608e-05, 9, 2, 1, 1)
0.0010093355895552118 (6.142992674295085e-05, 8, 0, 2, 3)
0.0010093355895552118 (6.142992674295085e-05, 8, 0, 3, 2)
0.0025233389738880297 (0.0001535748168573771, 8, 1, 1, 3)
0.0025233389738880297 (0.0001535748168573771, 8, 1, 3, 1)
0.0032442929664274664 (0.00019745333595948486, 8, 3, 1, 1)
0.0043257239552366225 (0.0002632711146126465, 8, 1, 2, 2)
0.004710232751257656 (0.00028667299146710397, 7, 0, 3, 3)
0.0048664394496412 (0.0002961800039392273, 8, 2, 1, 2)
0.0048664394496412 (0.0002961800039392273, 8, 2, 2, 1)
0.011355025382496133 (0.000691086675858197, 7, 4, 1, 1)
0.0190764426425935 (0.0011610256154417709, 6, 5, 1, 1)
0.020186711791104237 (0.001228598534859017, 7, 1, 2, 3)
0.020186711791104237 (0.001228598534859017, 7, 1, 3, 2)
0.022710050764992266 (0.001382173351716394, 7, 2, 1, 3)
0.022710050764992266 (0.001382173351716394, 7, 2, 3, 1)
0.02595434373141973 (0.001579626687675879, 7, 3, 1, 2)
0.02595434373141973 (0.001579626687675879, 7, 3, 2, 1)
0.0389315155971296 (0.0023694400315138185, 7, 2, 2, 2)
0.06358814214197835 (0.0038700853848059034, 6, 4, 1, 2)
0.06358814214197835 (0.0038700853848059034, 6, 4, 2, 1)
0.06594325851760717 (0.004013421880539455, 6, 1, 3, 3)
0.08478418952263779 (0.005160113846407871, 6, 3, 1, 3)
0.08478418952263779 (0.005160113846407871, 6, 3, 3, 1)
0.1271762842839567 (0.007740170769611807, 6, 2, 2, 3)
0.1271762842839567 (0.007740170769611807, 6, 2, 3, 2)
0.1453443248959505 (0.008845909450984922, 6, 3, 2, 2)