[arw]

Questions

1) Will hitting 2 pair win at showdown?

2) Will hitting 3 of a kind win at showdown?

Assumptions:

The pot is 100 chips.

There are 2 cards to come (turn + river)

The opponent is betting B chips.

You would be calling B to win (100 + B).

Calling EV = (%call)[(%win)(amount win) – (%lose)(amount lose)]

To profit from calling, your CallEV >= 0

To do the math, I’m going to do it from the villains perspective to find out the minimum bet size to have callEV > 0. This should be equivalent to what you want.

Case 1 — only gutshot outs

4 outs

CallEV = 0 = C * (W * (P + B) – L * (B))

%call

You’re assuming that you will call the bet every time, thus C = 1 or 100%.

%win

To win, you need to hit 4 outs of 47 cards and then assume that you don’t improve by hitting 43 non-outs on the 46 remaining river cards. This looks like, (4/47)*(43/46) = 8%….to hit 4 outs on turn or river.

0 = (8%)(100+B) – (92%)(B)

0 = B(8% – 92%) + (8%)(100)

B(84%) = (8%)(100)

B = 10 chips.

If the opponent bets less than 10 chips, it’s a profitable call with your gut-shot.

Case 2 — gutshot + 2 pair + trip outs

9 outs

CallEV = 0 = C * (W * (P + B) – L * (B))

%win

To win, you need to hit 9 outs of 47 cards and then assume that you don’t improve by hitting 38 non-outs on the 46 remaining river cards. This looks like, (9/47)*(38/46) = 16%.

0 = (16%)(100+B) – (84%)(B)

0 = B(16% – 84%) + (16%)(100)

B(68%) = (16%)(100)

B = 24 chips.

If the opponent bets less than 24 chips, it’s a profitable call with your gut-shot.

[eanderson85]

So, you are saying that bottom pair + gs draw on the flop mathematically shouldn’t call a 1/4 pot bet?
This is where position, stack sizes, reading your opponent, and determining implied odds is important, but which, if any, determines a call? And how big of one?