TD 2-7: discounting outs

[MarkGritter]

One of the great things about triple draw is that you and your opponents are often going for the same cards. How do you account for the probability that some of your outs are dead?

The easiest situation to analyze using probability alone is when your sole opponent is pat on the first draw. Most opponents need an 8 or better, although some will stand pat with a 9 as well. Additionally, the cards you hold affect the distribution of what Villian can hold. So I came up with a quick spreadsheet that:

1) Based on your hand, calculates the relative probability of each pat hand (for an opponent with an 8 or better and with a 9 or better.)
2) Calculates the probability, weighted by the hand distribution, that an out is in your opponent's hand.

Some sample results:

You hold 2345.
Villian is pat with 8 or better: 75% probability he has a 7 and 80% probability that he holds an 8. With 9 or better: 64% probability that he has a 7, and 65% that he holds an 8.

You hold 2367.
8 or better: 71% he holds a 4, 71% he holds a 5, 81% he has an 8.
9 or better: 67% he holds a 4, 65% he holds a 5, 65% he holds an 8.

You hold 3467.
8 or better: 81% he holds a 2, 72% he holds a 5, 81% he holds an 8.
9 or better: 68% he holds a 2, 65% he holds a 5, 66% he holds an 8.

You hold 2237:
8 or better: 70% he holds a 4, same for 5 and 6, 83% he holds an 8.
9 or better: 68% he holds a 4, 65% he holds a 5, same for 6 and 8.

For discounting outs, it's sufficient to just add the partial outs of each possibility, rather than calculate the probability of each case occuring. (For example, with 68% a 4 is held and 65% a 5 is held, you are missing about 1.33 outs.)

Can we come up with a rule of thumb here? One idea is to discount about 3/4 of an out for each rank. (For example, with 2345x you have about 6.5 discounted outs.)

But, you might make your hand and still lose. If you hold 2345x, then vs. an opponent with an 8 or better you have only 5.5 outs on average (since you will lose with an 8 or tie some of the time.) Against a possible 9 you pick up some extra equity.

With 2347x you are in better shape, with an average of 6.7 outs to a winning 7, about what the rule of thumb predicts, but in addition the possibility of a winning or tying 8 is worth 1.5 outs. Thus your 12 nominal outs have to be discounted by 3.8 against a likely 8.

With 2348, you have only about 6.2 outs against the range of 8s or better. This is a huge discount from 12. Still, you are better off than with 2345!

I can send the spreadsheet to anyone who is interested; PM me.