6 Outs
Semi-Bluffing
1) Our opponent folds (40 percent), we win the pot of $30.
2) Our opponent calls (60 percent), we hit our
draw (6/46), we win the $30 pot and the $25
call.
3) Our opponent calls (60 percent), we miss our
draw (40/46), we lose our $25 bet.
<Semi-
Bluff> = (0.4)(30) + (0.6)(6/46)(30+25) + (0.6)(40/46)(-25)
<Semi-
Bluff> = 12 + 4.3 - 13.04
<Semi-
Bluff> = $3.26
Checking
1 )Miss our
draw (40/46), no win or loss.
2) Hit our
draw (6/46), no extra money goes in (95 percent), win the pot of $30.
3) Hit our
draw (6/46), get stacks in (5 percent), win the pot of $30 and the $25 bet.
<
Check> = (40/46)(0) + (6/46)(0.95)(30) + (6/46)(0.05)(25+30)
<
Check> = 0 + 3.72 + 0.36
<
Check> = $4.08
This shows that the more
outs that we have to the
nuts the better our EV for both checking and semi-bluffing assuming all other variables stay the same. Whilst checking is still the better
option in this scenario the EV of semi-bluffing has increase by more than the EV of checking. So I would assume that there is a tipping point where <Semi-Bluffing> is better than <Checking>.
8 Outs
Semi-Bluffing
1) Our opponent folds (40 percent), we win the pot of $30.
2) Our opponent calls (60 percent), we hit our
draw (8/46), we win the $30 pot and the $25
call.
3) Our opponent calls (60 percent), we miss our
draw (38/46), we lose our $25 bet.
<Semi-
Bluff> = (0.4)(30) + (0.6)(8/46)(30+25) + (0.6)(38/46)(-25)
<Semi-
Bluff> = 12 + 5.74 + 12.39
<Semi-
Bluff> = $5.35
Checking
1 )Miss our
draw (38/46), no win or loss.
2) Hit our
draw (8/46), no extra money goes in (95 percent), win the pot of $30.
3) Hit our
draw (8/46), get stacks in (5 percent), win the pot of $30 and the $25 bet.
<
Check> = (38/46)(0) + (8/46)(0.95)(30) + (8/46)(0.05)(25+30)
<
Check> = 0 + 4.96 + 0.48
<
Check> = $5.43
Once again we see that the EV of Semi-bluffing has increased more than that of checking but has still not caught up to the EV of checking.
Extra Credit
If x is the % of the time we hit the
nuts
<Semi-
Bluff> = (0.4)(30) + (0.6)(x)(30+25) + (0.6)(1-x)(-25)
<Semi-
Bluff> = 12 + 33x - 15 + 15x
<Semi-
Bluff> = 48x - 3
<
Check> = x[(0.95)(30) + (0.05)(55)]
<
Check> = x[28.5 + 2.75]
<
Check> = 31.25x
<Semi-
Bluff> = <
Check>
48x - 3 = 31.25x
192x - 12 = 125x
67x = 12
x = 12/67 (0.179)
8/46 = 0.1739
9/46 = 0.1957
So the point at which semi-bluffing becomes better than checking in this scenario is when we have 9
outs to the
nuts.
Assuming I've not made a silly mistake.