Let's take the case of KK vs AA allin after a blank flop. After the flop, there are 45 cards left. If we run it once:

EV for KK = Pr(K on turn and no A on river) + Pr(no A on turn and K on river) + Pr(K's on turn and river) = 2/45 * 41/44 + 41/45 * 2/44 + 2/45 * 1/44 = 8.383838...%

Notice that the first two terms are the same because turn/river is interchangeable. Double checking this on pokerstove and using a flop with 0 chances of runner runner flush/straights, we get 8.384%. Nice. Exact.

Let's say we run it the second time. A couple possibilities in the first run:

- one A came out (2/45 * 41/44 * 2 = 8.2828%)
- then EV for second run is 2/43 * 40/42 * 2 + 2/43 * 1/42 = 8.9701%
- two A's came out (2/45 * 1/44 = 0.1010%)
- then EV for second run is 2/43 * 2 - 2/43 * 1/42 == 2/43 * 41/42 * 2 + 2/43*1/42 == 9.1915%
- one K and one A came out (2/45 * 2/44 *2 = 0.4040%)
- then EV for second run is 1/43 * 41/42 * 2 = 4.5404%
- one K came out and no A's came out (2/45 * 41/44 *2 = 8.2828%)
- then EV for second run is 1/43 * 40/42 * 2 = 4.4297%
- two K's came out (2/45 * 1/44 = 0.1010%)
- then EV for second run is 0%
- no A/K came out (41/45 * 40/44 * 2 )
- then EV for second run is 2/43 * 39/42 * 2 + 2/43 * 1/42 = 8.7486%

The above EVs were also double checked with pokerstove using dead cards (A, K and blanks) and should be the exact probabilities. Adding these all up, the EV over all cases for the second run is 8.3838%- same as the first run.