I know you’re using hyperbole, but I’m going to do the calculations anyway :)
If you bet a fraction x of the pot, with prob p of winning, and no outs, then your EV is p-(1-p)x. Clearly, EV>0 for optimal play, and a half-pot sized bet is common, so p-(1-p)/2>0 ⇒ p>1/3.
So the bluff should succeed at least 1⁄3 of the time.
Now suppose I have made some large bets, and you think I have at least JJ with 95% prob, and am bluffing with junk with 5% prob. I think you can beat JJ with 30% probability. I might chose to bet half the pot with all my possible hands (I’m now playing a probability distribution, not a hand), in which case you have to fold with 70% of your hands because 0.05 (1+0.5)<0.95 1. So in this case, my bluff succeeds 70% of the time, with EV 0.7-(1-0.7)/2=0.55.
Of course this is a massively simplified example.
Apparently, according to a book I read, if two pros playing head up no-limit are dealt 9 4, the author estimated that the person who has position (plays second) has around 2⁄3 chance of winning by bluffing his opponent off the hand, and of course the person who plays first might win by bluffing as well. So this seems to indicate that there is a reasonable chance to win by bluffing.
Overall, I think pros don’t make so many dramatic all-in bluffs, and in fact tend to semi-bluff, by betting with hands that have outs anyway.
I know you’re using hyperbole, but I’m going to do the calculations anyway :) If you bet a fraction x of the pot, with prob p of winning, and no outs, then your EV is p-(1-p)x. Clearly, EV>0 for optimal play, and a half-pot sized bet is common, so p-(1-p)/2>0 ⇒ p>1/3.
So the bluff should succeed at least 1⁄3 of the time.
Now suppose I have made some large bets, and you think I have at least JJ with 95% prob, and am bluffing with junk with 5% prob. I think you can beat JJ with 30% probability. I might chose to bet half the pot with all my possible hands (I’m now playing a probability distribution, not a hand), in which case you have to fold with 70% of your hands because 0.05 (1+0.5)<0.95 1. So in this case, my bluff succeeds 70% of the time, with EV 0.7-(1-0.7)/2=0.55.
Of course this is a massively simplified example.
Apparently, according to a book I read, if two pros playing head up no-limit are dealt 9 4, the author estimated that the person who has position (plays second) has around 2⁄3 chance of winning by bluffing his opponent off the hand, and of course the person who plays first might win by bluffing as well. So this seems to indicate that there is a reasonable chance to win by bluffing.
Overall, I think pros don’t make so many dramatic all-in bluffs, and in fact tend to semi-bluff, by betting with hands that have outs anyway.