Pots odds versus bluff rate to call on the river

You get to the river and you have nothing but a bluff catcher. Bottom pair on a scary board. If they made their hand they have they have you beat. But you can beat on outright bluff.

But you are getting 3:1 to call
How often does your opponent need to be bluffing to call

This question seems difficult but when you start knowing what it means, things become easier.

When you say pot odds are 3:1 to call it means that in this situation you only need to win 1 time out of 4. Let's put on example, this pot odds occur when villain bets half pot:

Let's say on the river pot is 2€ and villain bets 1€. When you lose on showdown, you lose 1€ (your call), but when you win, you earn 3€ (his bet and the pot).

• 1# lose 1€
• 2# lose 1€
• 3# lose 1€
• 4# win 3€

So, what is happening? In this situation no matter what if you win 25% of the times, the EV will be 0.

Thus, the answer of your question is, if he is bluffing more than 25% of the times you can call him with all bluffcatchers

Always try to think what are meaning the pot odds and you will think for your own

b is bluff rate

EV = 0 = b*3 - (1-b)*1
0 = 4b - 1
b = 1/4

bluff rate is always 1 / (pot odds + 1)

what if they bet 5x pot
pot odds (1 + 5)/5 = 6/5
EV = 0 = b*6 - (1-b)*5
0 = 11b - 5
b = 5 / 11 1 / (6/5 + 1) = 1 / (11/5) = 5 / 11

what if they bet 100x pot
pot odds (1 + 100)/100 = 101/100
EV = 0 = b*101 - (1-b)*100
0 = 201b - 100
b = 100 / 101 ~ 1/2
The math does not change that much from 5x pot bet to a 100x