I'm creating AI for a card game, and I run into problem calculating the probability of passing/failing the hand when AI needs to start the hand. Cards are A, K, Q, J, 10, 9, 8, 7 (with A being the strongest) and AI needs to play to not take the hand.
Assuming there are 4 cards of the suit left in the game and one is in AI's hand, I need to calculate probability that one of the other players would take the hand. Here's an example:
AI player has: J Other 2 players have: A, K, 7
If a single opponent has AK7 then AI would lose. However, if one of the players has A or K without 7, AI would survive. Now, looking at possible distribution, I have:
P1 P2 AI --- --- --- AK7 loses AK 7 survives A7 K survives K7 A survives A 7K survives K 7A survives 7 KA survives AK7 loses
Looking at this, it seems that there is 75% chance of survival.
However, I skipped the permutations that mirror the ones from above. It should be the same, but somehow when I write them all down, it seems that chance is only 50%:
P1 P2 AI --- --- --- AK7 loses A7K loses K7A loses KA7 loses 7AK loses 7KA loses AK 7 survives A7 K survives K7 A survives KA 7 survives 7A K survives 7K A survives A K7 survives A 7K survives K 7A survives K A7 survives 7 AK survives 7 KA survives AK7 loses A7K loses K7A loses KA7 loses 7AK loses 7KA loses
12 loses, 12 survivals = 50% chance. Obviously, it should be the same (shouldn't it?) and I'm missing something in one of the ways to calculate.
Which one is correct?