Using emailed hand histories, I tabulated the results of 1675 pots in

PokerStars tournaments in which betting ended prior to the river because

players in the pot were all-in, and in which none of the all-in players was

drawing dead. The great majority of the tournaments were multi-table; a few

were two-table, and fewer still single-table. There were a total of 3420

all-in hands (usually two per pot, but in a few instances three or even

four).

For each all-in, I collected the player's equity in the pot at the time of

the all-in and the proportion of the pot the player won (1.0 for a win, 0.0

for a loss, 0.5 for a two-way split, etc.) The pot equity of a hand is the

average proportion of the pot that a player would win over the long run.

For example, in the following sample Hold 'Em Showdown output,

990 pots with board cards: Tc 9d 7s

9h9s 8d8s

% chance of outright win 74.040404 24.343434

% chance of win or split 75.656566 25.959596

expected return, % of pot 74.848485 25.151515

fair pot odds:1 0.336032 2.975904

pots won: 741.00 249.00

--the "expected return, % of pot" result shows the pot equity of each hand.

The chi square statistic does not indicate significant variance of actual

from expected (3392.7 on 3421 hands). A plot of rolling 30-"bin" averages

of sorted equity vs. pots won looks linear; the linear regression is

y = 0.9908x + 0.0045

which is not significantly different from the theoretical y = 1.0x + 0 that

describes the "no dealing bias" case.

Looking at the "big" favorites/underdogs, and expressing the favorite's

chances as odds against the underdog:

Odds at least # of hands Avg equity Avg pot won

2:1 1121 0.8123 0.8134

3:1 753 0.8617 0.8597

4:1 623 0.8806 0.8777

9:1 260 0.9383 0.9321

In sum, I see no evidence that "bad beats" are more frequent than expected.

(I am grateful to Paul Pudaite for analytical guidance, but any errors are

my own.)

--

For mail, please use my surname where indicated:

st...@surname.reno.nv.us (Steve Brecher)