The second table is identical to the first table but is slightly more readable and rounds up to the nearest integer (in most cases). For example,
if you start with five trumps, there is a 10% chance that your partner will have exactly one of the others. That can be found by looking in the Trumps
to Start column, going down to "5", then across that row to the column that says "Probability Partner has 2". The last column is helpful if you are
wanting/needing your partner to have at least one of the trumps you don't have. For example, if you start with four trumps, your partner will have
one or more of them 72.6% of the time, which is almost 3 out of 4 times.
| Trumps to Start | Probability Partner has none | Probability Partner has 1 | Probability Partner has 2 | Probability Partner has 3 | Probability Partner has 4 | Probability Partner has 5 | Probability Partner has 6 | Probability Partner has the rest | Probability Partner has 1 or more |
| 0 | 0.029515 | 0.180779 | 0.361558 | 0.301299 | 0.109563 | 0.016434 | 0.000843 | 0.000009 | 0.970485 |
| 1 | 0.055341 | 0.258256 | 0.387384 | 0.234778 | 0.058695 | 0.005418 | 0.000129 | 0.000129 | 0.944659 |
| 2 | 0.098383 | 0.344341 | 0.375645 | 0.156519 | 0.02408 | 0.001032 | 0 | 0.001032 | 0.901617 |
| 3 | 0.167251 | 0.425731 | 0.319298 | 0.081871 | 0.005848 | 0 | 0 | 0.005848 | 0.832749 |
| 4 | 0.273684 | 0.478947 | 0.221053 | 0.026316 | 0 | 0 | 0 | 0.026316 | 0.726316 |
| 5 | 0.433333 | 0.466667 | 0.1 | 0 | 0 | 0 | 0 | 0.1 | 0.566667 |
| 6 | 0.666667 | 0.333333 | 0 | 0 | 0 | 0 | 0 | 0.333333 | 0.333333 |
| 7 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Trumps to Start | Probability Partner has none | Probability Partner has 1 | Probability Partner has 2 | Probability Partner has 3 | Probability Partner has 4 | Probability Partner has 5 | Probability Partner has 6 | Probability Partner has the rest | Probability Partner has 1 or more |
| 0 | 3% | 18% | 36% | 30% | 11% | 2% | 0.08% | 0.00% | 97% |
| 1 | 6% | 26% | 39% | 23% | 6% | 1% | 0.01% | 0.01% | 94% |
| 2 | 10% | 34% | 38% | 16% | 2% | 0.10% | 0.10% | 90% | |
| 3 | 17% | 43% | 32% | 8% | 1% | 0.59% | 83% | ||
| 4 | 27% | 48% | 22% | 3% | 3% | 73% | |||
| 5 | 43% | 47% | 10% | 10% | 57% | ||||
| 6 | 67% | 33% | 33% | 33% | |||||
| 7 | 100% | 0% | 0% |
For those who are interested in how the actual calculations are done, let's look at a couple of the entries in this table.
Let's say I am starting with four trumps. To calculate the probability that my partner has exactly one of them I have to count the number of hands
that will give that result. There are three remaining trumps and my partner can choose only one of them - That's C(3,1). Then we must fill his/her
hand with six more dominoes out of the 18 others. We get to 18 as follows: 28 dominoes to start, but I have 7 leaving 21. But we have to take the
3 trumps out which leaves us with 18. Therefore the other 6 in the partner is represented by C(18,6), or 6 ne of the trumps and six other random
dominoes would then be C(3,1)*C(18,6) which is equal to 55,692. The total number of possible hands that my partner might have is C(21,7), or
"21 choose 7". Again, that's because he/she has 21 dominoes from which to draw 7. C(21,7) is equal to 116,280.
At this point it's a simple division: (Number of hands containing exactly one trump)/(Number of possible hands). 55,692/116,280 gives us the 48%
probability (rounded up from 47.8947).
In Excel, that one cell would look like this: =COMBIN(3,1)*COMBIN(18,6)/COMBIN(21,7)
Similarly, the cell representing the probability that my partner has two of the other three trumps in Excel would be:
=COMBIN(3,2)*COMBIN(18,5)/COMBIN(21,7). Notice how it's "18 choose 5" instead of "18 choose 6" now because he already has
two (of the three) trumps. That formula yields the 22% value. The other entries in the table are all derived in a similar manner.
If I start with six trumps, it is obviously intuitive that there is a 33.33% chance of my partner having the remaining one.
That's indeed the case and also holds true for ANY particular domino that you don't have. For instance, I don't have the 5:5,
so there's only about a 33% chance my partner does. And conversely a 67% chance one of my opponents does. But we can use the
same type of calculation using Combinatorics which both verifies that and shows the authenticity of the method. The calculation
for that would be C(20,6)/C(21,7); i.e. "20, choose 6" divided by "21, choose 7". That's because if my partner has the "needed
domino" (the other trump, or the 5:5 as mentioned), there are then 20 dominoes to choose the remaining 6. An Excel entry with
=COMBIN(20,6)/COMBIN(21,7) does indeed result in 33.33%.
The other entries in the table are all derived in a similar manner.
