[Mike Haven]

i've now done my math on this and i was wrong, previously, in "actual dollars won" terms:

if hero calls then 308 (7 x 44) times out of 1980 (45 x 44) hero will lose 300 = 92,400 (assuming hero folds if heart turns)

if hero bets non-flush turn all in then 1672 (38 x 44) times out of 1980 he will win the 300 = 501,600 (assuming villain folds)

net win = +409,200 or +207 per hand

if hero raises all in then 308 (7 x 44) times out of 1980 (45 x 44) hero will lose 900 = 277,200 on turn (assuming villain calls)

and 266 (7 x 38) times out of 1980 hero will lose 900 = 239,400 on river

and 1406 (38 x 37) times out of 1980 he will win 900 = 1,265,400

net win = +748,800 = +378 per hand

therefore it is "better" to push all in on the flop

my previous answer seems to give the "right" answer if you look at "dollars won per dollar staked":

you win $207 for a stake in play of $300

or you win $378 for a stake in play of $900

i suppose the yardstick of poker is "actual dollars won"

however, if you didn't want to run the risk of losing a number of $900 stakes before the "long run" cut in, then winning $171 less for stakes of only a third the size of $900 might still be regarded as the prudent way to go, for those on a limited bankroll