payoff probability

On the river:
$100 pot Opponent bets $20 I can only beat a bluff.

I have to think he's bluffing what percentage of the time to make a call profitable? What are my calculations?

Thanks in advance

if you don't know how to solve for x here, it may help to fix x and calculate your expectation. you will then use the same equation with the expectation fixed which will allow you to calculate x.

assume your opponent is bluffing 10% of the time.
your value for a call is 90% of -$20 plus 10% of $120, or:
(0.9 * -20) + (0.1 * 120) = -18 + 12 = -6.
in words, if your opponent is bluffing 10% of the time and you call, you will lose your bet 9 times out of ten and win the pot 1 time out of 10. your expectation in this case is thus -$6.

if that is how you solve for your expectation, you can set your expectation to 0, the break-even point, and now solve for the bluff percentage x instead:
((1-x) * -20) + (x * 120) = 0
note x here is actually just a real number; mulitply the answer by 100 to express it as a percentage.

that should allow you to do the calculation yourself. note that if your answer comes out to less than 10%, you have made a mistake, as you already know that your expectation in that case is negative.

good luck
The Club

[Homer]

You're getting 6:1 on your call. If the odds that he is bluffing is better than 6:1 against, then you should call. 6:1 is 1/7, which is about 14%. So if you think there is greater than a 14% chance that he is bluffing you should call.

-- Homer