Originally Posted by LawDude

No the EV is not the same as the odds of winning the second hand will change as the first turn and river are delt (if your all in on the flop). So you won't be expected to win (Expected Value) the same in both board 1 and board 2 as the equity will have changed.

It will change, however, it will change in one direction if you hit the first board, and the other direction if you miss. And the rate of change in each direction will be proportionate to the likelihood of you hitting the first board.

Which makes the total change in EV zero.

Here's a simple example. You have AA, villain has JJ, and the board is J972 rainbow. You have 2 outs on the river.

If you run it once, your equity is 2/44th (1/22nd) of the pot.

If you run it twice, your equity the first go around is 1/2 of 2/44th of the pot, or 1/44th.

Your equity the second go around? Well, the 42/44ths of the time you miss the first board, your equity will be 1/2 of 2/43rd of the second pot, or 2/86th of it.

The 2/44th of the time that you hit the first board, your equity will be 1/2 of 1/43rd of the second pot, or 1/86th of it.

Now, let's work out the math:

Running it once:

Equity = 1/22 = 4.55 percent

Running it twice

Equity = 1/44 + ((21/22) x (1/43)) + ((1/22) x (1/86)) = 4.55 percent