Probabilities of not getting set on trumps on an 84 bid
One of the most exciting events during a game of 42 is bidding two marks (84), or sometimes
even three or four marks. You can have a laydown 84 hand, where no one can set you
(assuming you play it correctly of course). You might however have a strong hand after your
trumps, but you may not be guaranteed to pull all the trumps that you need with the ones you
have. This page will provide the probabilities of making 84 only from that "trump standpoint".
But these numbers can also be used in general to make good decisions based solely on the
trumps that you start with, even if it's not an 84 attempt.
This will not contain any of the situations where you will be able to pull all the trumps, as by
definition those are all 100%. For instance, if I start with four trumps and have the top three of
them, I can use all three if necessary to pull the others even if I'm "three trumped" by an
opponent. These all also begin with the fact that you have the double. If you don't, that's an
easy 33% calculation of making 84, since your partner would have to have it.
Note also that these probabilities are for not being set by an opponent via trumps. They include
the cases where your partner might be able to have the high one at some point. In other words,
it;s not that you will necessarily get them all in, but only that you will get the proper ones from
your opponents. For example, I might have four trumps with the top two. I am set if an
opponent has the other three, but good to go if my partner does. Naturally, if your partner is
able to help out with trumps, that's probably always a winning 84 hand.
For notation purposes, T1 is "Trump 1" aka the "Double" of your trumps (or also the "bull")
T2 is the next highest, aka "the cow". T3 is the third highest ("the calf"), and on down the
line to T7 as the lowest.
So, if trumps are aces, T1=1/1, T2=1/6, T3=1/5, T4=1/4, T5=1/3, T6=1/2 and T7 =1/0.
They are listed in descending order from highest to lowest probability.
Probabilities to make 84 in regard to trumps:
(Top three only)
T1 T2 T3 0.988304 99% (only being four-trumped will hurt you)
(Top two plus any other two other than T3)
T1 T2 T4 T5,
T1 T2 T5 T6, etc. 0.947368 95% (one opponent must have all other three)
(Top two plus T4)
T1 T2 T4 0.865497 86%, or 13 in 15 (one opponent must have three including T3, or all four others)
(Top two without T3 or T4)
T1 T2 T5
T1 T2 T6, etc. 0.824561 82.5%, or 9 in 11 (one opponent must have you three- or four-trumped)
(Any five with the double)
T1 T3 T4 T5 T6
T1 T4 T5 T6 T7, etc. 0.800000 80%, or 4 in 5 (one opponent must have other two)
(Four but without the cow T2)
T1 T3 T4 T5
T1 T3 T5 T6, etc. 0.652632 65.2%, or nearly 2 in 3
(Top two trumps only)
T1 T2 0.636739 63.7%, or 7 in 11
(Three without T2, but with T3 & T4)
T1 T3 T4 0.546199 54.6%, or 6 in 11
(Four without T2 or T3)
T1 T4 T5 T6
T1 T5 T6 T7 0.505263 50.5%, or just slightly better than a coin flip
(Three without T2, but with T3)
T1 T3 T5
T1 T3 T6, etc 0.505263 50.5%, or just slightly better than a coin flip
(Three without T2 or T3, but with T4)
T1 T4 T5 0.292398 29.2%, or just under 3 in 10, or "not very good"
(Three without T2, T3 or T4)
T1 T5 T6
T1 T6 T7 0.185965 18.6%, or "even worse"
Takeaways from this - having T2 (or "the cow") in addition to the double obviously yields the
highest probabilities. However, it till possible without it, especially if you have five trumps to
start with. Only 20% of the time will one of your opponents have the other two. And having only
four without T2, but having T3 is still a respectable 65.2%, with having top two only right behind
that at almost 64%.
Having this information, use the score of the game to maybe take some chances you might not,
as an overbid or whatever.