Trading card game poss

External question — Cross Validated Stack Exchange Author: Patrick Original post: https://stats.stackexchange.com/questions/643002 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. A trading card game that is called Flesh and Blood (description and rules here ) has two players construct a 60 card deck and the hand limit is 4. Player 2's Deck color combination: 60 cards in deck; 34 are blue, 17 are red, and 9 are yellow. Statistical scenario: Player 1 is at 26 Health Points and has 1 card in hand. Player 2 has 4 cards in hand(3 blues and 1 yellow) and 1 card(red) face down on the table. Player 2 can only win if the top card of their deck (Y) and card in Player 1's hand is (X): X(P1)/Y(P2): red/red = Win red/yellow = Win red/blue = Win yellow/red = Lose yellow/yellow = Win yellow/blue = Win blue/red = Lose blue/yellow = Lose blue/blue = Lose Winning Scenarios/Total Scenarios = 5/9 or 55.6% to win. My question is: Do I need to add the percentage of what colors are left in Player 2's deck when the scenario above is conducted? During this scenario, Player 2's deck is at 45 cards. 4(3 blue and 1 yellow) of them in hand, 1(a red) on table, and 10 in Graveyard(4 reds, 3 yellows, and 3 blues). Player 2 adds -1 to every red and yellow and +1 to every blue. With the combination of hand and graveyard color count, Player 2 will be at a Blackjack Count of -3, showing a slight chance to see a blue over reds and yellows. 34 blues/60 deck size = .5667 or 56.67% to have a blue be the top of the deck at the start of the game. -1/+1 to the total count of blues in the deck, shows it has a difference of +/-1.66%. When I am at a count of -3, I do this: 1.66x3= 4.98%. So an increase of 5% to see a blue on top 2 of the 3 blue combinations, Player 2 wins, so 67% chance to win if its a blue on top. Do I add the 5% to the win probability? Let me know if this is too confusing or not.

External answer — Cross Validated Stack Exchange Author: Sextus Empiricus Original post: https://stats.stackexchange.com/a/643006 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. Using these black jack counts makes it too complicated. Player 2 can accurately compute the composition of the remaining cards in the deck. $$\begin{array}{} \text{red} &=& 17 - (4+0+1) &=& 12 \\ \text{yellow} &=& 9 - (3+1+0) &=& 5 \\ \text{blue} &=& 34 - (3+3+0) &=& 28 \\ \end{array}$$ If each of the 45 cards has equal probability to be a top card then the probability for a blue card on top is 28/45 ≈ 62.2%.

External answer — Cross Validated Stack Exchange Author: JadeImp Original post: https://stats.stackexchange.com/a/672293 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I’d say yes you should factor in the updated deck composition, because the probabilities of drawing each color are no longer uniform once you account for cards in hand, graveyard, and the running count. That “+5%” adjustment for seeing a blue on top reflects the real shift in likelihood due to what’s already been played, so adding it to your win probability gives a more accurate estimate of Player 2’s chance to win in this scenario. It’s a small tweak, but in a game with tight margins like Flesh and Blood, every percentage point can matter.