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%.
Interestingly, I ran these numbers a while back in regards to leading a cold 5 off.
If you consider only the 5-5 & 5-0, you have nearly a 70% chance of making 31 by leading the 5-4 first.
That's 33.3% that your partner has the double plus 20.0% that either opponent has both,
plus 16.4 % that your partner has 5-0 but another to protect it. If all three other counters
were available for an opponent to hold one or more of, have no fives, and the other opponent
with the 5-5, that goes down to 62%. But in this case, the 6-4 is in your hand, so it would be
somewhere between the 62 and 70%
