How dead are my outs? A surprising TD 2-7 result.

[MarkGritter]

Conventional wisdom is that 7's and 8's are more dead than other outs. But against opponents who are playing tightly they are most likely to be live!

Method: Enumerate and count all 2-7 starting hands listed in SS2 as playable. (This list is somewhat more tight than I, for example, play.) This includes any pat 7 or 8, any 1-card 7 draw which includes a 2 (including 2345x, 2456x, etc.) any 1-card 8 draw which does not include a straight (2348 through 3678), any two-card draw to a seven which includes a deuce, and any two-card draw to #5. (Notably this list does not include 268xx, 278xx, 358xx, etc.)

Count the number of possible ways to make each hand, including those with pairs. Weight the number of outs consumed by each hand by its probability. Total number of possibilities considered is 433584 hands, 15% of the five-card hands. For example, there are 1024 ways to make 75432--- although four of them are flushes. This contributes a 'weight' of 1024/433584 applied to each of 2, 3, 4, 5, and 7. There are 144 ways to make 77332, which means a weight of 144/433584 applies to 2, and 288/433584 to 7 and 3.

Result: Weighted number of 'dead' outs by card:

2: 0.985313
3: 0.53275
4: 0.481678
5: 0.481678
6: 0.434407
7: 0.434407
8: 0.437322

That is, any player in the pot is almost certainly going to have a deuce. But, on average they hold only 0.44 eights. (Not quite a 44% chance, since sometimes they have two or three.)

Thus, in the absence of other information, deuces are the most dead (confirming conventional wisdom). Threes, fours, and fives are all about equally likely to be dead (about half an out each.) But sixes, sevens, and eights are actually the least likely to be held by another player.

This analysis is incomplete, however, because the contents of a player's hand alters the probabilities. For example, if Hero holds 2345x then the other players are slightly more likely to holds 7s and 8s. If Hero holds 2347 then 6s and 5s are more likely in other player's hands.

The analysis is also incomplete because players may give clues about their holdings, or play a wider range of hands. Adding junk hands like 754, 864, 37, etc. may also alter the results.

My intuition is that multiple holdings will tend to smooth out the expected number of dead outs rather than alter them, but I could be mistaken.

The detailed calculation, in the form of an Excel spreadsheet, is available on request.