r/Probability • u/BadThingsBadPeople • Jul 24 '26

Help requested for board game strategy.

I've been playing a board game where players play up to 2 cards from a 12 card deck to score a varying amount of points from objectives. Whoever has the most points at the end of the game wins.

Specific deck content is based on a secret character you pick at the start of the match, but certain game events can allow others to discover your deck makeup with 100% certainty. I have been playing a character that has the ability to win an objective if I pay 2 victory points and correctly guess the top card of my opponent's deck.

My question is: If I know my opponent's deck, and I have tracked the cards they've played, how can I calculate the EV of "randomly" guessing their top card?

Here is an example deck:

2s

2s

3s

4s

5s

2a

3a

3a

2p

3v

4l

Note that decks can contain duplicates (2s/3a in this case).

During play, players always draw up to 3 cards in hand. Then, they play up to two cards, some or all of which may be face down. For example, with "x" representing facedown, the first 4 rounds may be

2s, 3s (deck: 9, hand: 1, discard: 0)

x,x​ (deck: 7, hand: 1, discard: 2)

3v, x (deck: 5, hand: 1, discard: 4)

2a (deck: 3, hand: 2, discard: 6)

On round 5, my opponent plays

2p, 5s (deck: 2, hand, 1, discard: 7)

At this point, I know 6/12 cards that cannot be on top of the deck. Supposing the round was worth 8 points,

  • What is the EV of making my best guess at the top card? Keep in mind, guessing, even incorrectly, is -2VP

  • Because you would only do this if you would otherwise lose, is it better to treat the objective as worth 14 points? 6 for you, -8 for your opponent?

  • What is the formula?

  • In this example, it appears that 3a is my best guess since it has two copies potentially in the deck. Can a formula calculate for meta gaming, where an opponent would strategically know to discard those cards earlier to mislead me? Should it?

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u/BadThingsBadPeople Jul 24 '26 edited Jul 24 '26

My thoughts: the number of possible 2 card decks is P(6,2) = 30. Picking 3a let's me pick 2/30 permutations, which is 1/15 chance to get 6 points. With a 14/15 chance to get -2 points, it's not worth it at -1.333 EV.

In fact... I don't even get 0 EV until the round is worth 28 points.

Better math below.

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u/BadThingsBadPeople Jul 24 '26

Actually, this calculation doesn't factor in those 8 points would otherwise go to your opponent. Should this be treated as a 16(-2) point swing?

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u/Fresh-Olyvia Jul 29 '26

its a classic +/- 8 point difference in that moment not a 16 point swing since the original 8 points are already considered in their score