[W. Deranged]

Time for some algebra, kids:

Let the pot size be P, including villain's turn bet.

Let our probability of improving be i.

Let the probability we win unimproved on the end be u.

Elindaur has told us:

1 / (P + 1) < i

1 / (P + 2) < u

What we want to know is whether we have the odds to call down, namely our chance of improving plus our chance of holding up unimproved is worth the two more bets.

We know that our odds of winning are:

i + (1-i)u

We want to know whether:

2/ (P+3)

which represents the odds that the pot is laying us (the pot size going in + 1 bet from villain on the river + the 2 BB we put in divided by the 2 BB we put in), is smaller than our odds of improving:

[i + (1-i)u] = i + u - iu

Now, notice that:

2 / [P+3] < 2 / [P+2] < (1 / [P+2]) + (1 / [P+1]) < i + u

The very small random correction term, (-ui), doesn't make that much of a different, really. I could do a more full proof but it's not really that interesting.

The point is that if you have odds to call on the turn strictly to improve and the odds to call on the river strictly to snap off a bluff you have collective odds to call down.

So calling is better than folding. (Which is to say nothing of whether raising is better, which it often may be.)