Poker: running cards multiple times

Consider the following a quick exercise in combinatorics. We are investigating the effects of running cards twice. You can see a real life example here. It is known that the EV doesn't change when you run multiple times (but you lower your variance). Let us check this claim.

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.