How can I calculate the probability of being able to purchase a card in my custom die system?

Question

I'm trying to work out the Probability of reaching a total with a custom die system.

Each die in this system has 6 sides: 1 Sword, 2 Sword, 3 Sword, Political, Castle, and Wizard.

Each player starts off with 7 such dice. On their turn, they can purchase cards from a central Supply if they manage to roll a total specified on the card. An example would be the Priest card which requires a player to get 4 Swords, 1 Sword, 1 Political, and 1 Castle. A player would declare that they will try to purchase this card, then roll their dice. They would try to combine dice to meet the 4 swords requirement, and/or spend dice to meet any other requirement listed. The order the requirements are met in does not matter. After spending as many dice as the player wants to, with no minimum, the player would reroll all remaining dice not spent to meet requirements and try again if they had not yet met all requirements. If no requirements were met this round, a 1 die penalty is paid by the player. Dice cannot be reserved across rounds.

I am interested in finding a way to quickly calculate the odds of successfully being able to buy a card on a turn given the cost of that card.

An example of an unsuccessful purchase would be:

1) The player declares they wish to buy the Priest card

2) The player rolls 1 Sword, 1 Sword, Political, Political, 3 Swords, Castle, 2 Swords.

3) They choose to place 1 Sword to meet the 1 Sword requirement, 3 swords and the other 1 Sword to meet the 4 swords requirement, choose not to spend any of the politicals to meet the Political requirement (no reason, just cause) and reroll now 4 dice as three are placed on the Priest Card fulfilling requirements.

The next rolls are all swords, so the player loses a single Dice and rerolls 3.

They get 3 political and so take one to complete the political requirement leaving 2 to play.

They roll, getting a 3-Swords and a Wizard, and so lose a die.

They roll, getting a 2-Swords, and fail to complete the requirements.

There are other mechanics in play with cards that alter conditions but I just need help with this aspect.

Been trying to use AnyDice and not having any real successes as examples and tweaks to existing code gives errors.

Again, I'm trying to quickly calculate probabilities of being able to purchase a card on a single turn given its price.

I'm intending to use this mechanic first in a card game and then to scale it up for use as the mechanics behind an RPG system.

Answer 1

There is a dice game almost exactly like what you're describing called Farkle. Farkle is a game where you try to make certain combinations of dice and when you make a combo, the dice involved are "locked in" removed from your rolling pool. As long as you make at least one scoring roll and lock in that score, you can keep rolling any dice that have not been locked in yet. Instead of scores, your game has costs for cards, but this doesn't change the math.

Here are some good articles about scoring and probability for Farkle:

https://graciesdad.wordpress.com/2009/08/30/farkle-odds/

http://www.jonmutchler.com/FarkleFun2.html

The word "Farkle" refers to "failed to score on a roll". So when you see "probability to Farkle," they are talking about the probability of rolling dice in a way that nothing can be locked in, at which point the player must stop rolling. 1 - P(farkle) = probability to score and keep rolling.

All you would have to do is replace the table of possible scores with the costs associated with your cards and map each side of your game's dice to 1-6 (e.g. "1-3 Sword" is 1-3, Political = 4, Castle = 5, Wizard = 6). The differences you have to account for is that Farkle is played with 6 dice and your game is played with 7, and that in Farkle a single 1 or 5 is a scoring roll, where in your game, a roll that contributes to the cost of a card is a scoring roll; so your "P(farkle)" changes based on the card being rolled for.

If you need help converting a 6-dice game to a 7-dice game, that's a separate question but I can try to help with that, too. Alternatively (if it doesn't break your game), you can make your game a 6-dice game and port it directly, reducing the costs of all of your cards by 1 to account for less dice.

Answer 2

I've coded up an incomplete simulation of your rule set in Python. From my strategy, it looks like about an eighty percent chance of buying the Priest card with seven dice. I've tested the code on TiO so you can run it without installing Python locally.

Here's an example of the debug output:

Reqs: [Sword_4, Sword_1, Political, Castle]
Rolls: [Wizard, Sword_1, Sword_3, Sword_1, Castle, Wizard, Sword_1]
Matched: Castle
4 matched by 3, 1
Sword_1 Matched exactly
No matches with: [Wizard, Sword_1, Wizard, Sword_1], still need [Political]
Rerolling 4
Reqs: [Political]
Rolls: [Castle, Wizard, Sword_3, Castle]
Current Matches: [Castle, Sword_3, Sword_1, Sword_1]
No matches with: [Castle, Wizard, Sword_3, Castle], still need [Political]
Rerolling 3
Reqs: [Political]
Rolls: [Sword_2, Sword_1, Sword_3]
Current Matches: [Castle, Sword_3, Sword_1, Sword_1]
No matches with: [Sword_2, Sword_1, Sword_3], still need [Political]
Rerolling 1
Reqs: [Political]
Rolls: [Sword_3]
Current Matches: [Castle, Sword_3, Sword_1, Sword_1]
No matches with: [Sword_3], still need [Political]
Rerolling -2
Failed to acquire Priest remaining:
Chance to buy card Priest is about 78.39% after 10000 iterations

Here's a successful attempt:

Reqs: [Sword_4, Sword_1, Political, Castle]
Rolls: [Castle, Sword_1, Sword_1, Wizard, Sword_1, Sword_1, Sword_3]
Matched: Castle
4 matched by 3, 1
Sword_1 Matched exactly
No matches with: [Sword_1, Wizard, Sword_1, Sword_1], still need [Political]
Rerolling 4
Reqs: [Political]
Rolls: [Political, Sword_2, Castle, Sword_1]
Current Matches: [Castle, Sword_3, Sword_1, Sword_1]
Matched: Political
Unneeded: [Sword_2, Castle, Sword_1]
Successfully acquired Priest

You can define new cards by modifying the line:

priest = card("Priest", dice.Sword_4, dice.Sword_1, dice.Political, dice.Castle)

Apologies for the verbosity, I was making sure that the strategy was working properly.

One example where my implementation falls short is failing to match a Sword 4 with a Sword 2 and two Sword 1s. That it relatively trivial to add, but it's probably something you'd choose if the Sword 4 was your only remaining requirement.

Actually, I just reviewed your rules and I've implemented the penalties wrong. I've just given the first re-roll free and added an increasing penalty for each subsequent round. It is relatively simple to track the matches within a round and apply penalties then, I suspect it'll make things easier to recruit the Priest.

Let me know if this approach is interesting or useful and I'll fix it up or feel free to give it a go yourself.

Edit: Okay, it was gnawing at me so I fixed the implementation here. As suspected the chance to acquire a Priest increases, to about eighty-five percent. Here's a new log showing an example including rerolls:

Reqs: [Sword_4, Sword_1, Political, Castle]
Rolls: [Sword_1, Sword_1, Wizard, Sword_1, Wizard, Sword_2, Wizard]
Sword_1 Matched exactly
No matches with: [Sword_1, Sword_1, Wizard, Sword_1, Wizard, Sword_2, Wizard], still need [Sword_4, Political, Castle]
Rerolling 7
Reqs: [Sword_4, Political, Castle]
Rolls: [Castle, Castle, Wizard, Sword_2, Castle, Sword_2, Sword_1]
Current Matches: [Sword_1]
Matched: Castle
4 matched by 2, 2
No matches with: [Castle, Wizard, Castle, Sword_1], still need [Political]
Rerolling 4
Reqs: [Political]
Rolls: [Sword_3, Sword_1, Sword_1, Sword_2]
Current Matches: [Sword_1, Castle, Sword_2, Sword_2]
No matches with: [Sword_3, Sword_1, Sword_1, Sword_2], still need [Political]
Rerolling 3
Reqs: [Political]
Rolls: [Sword_2, Sword_3, Political]
Current Matches: [Sword_1, Castle, Sword_2, Sword_2]
Matched: Political
Unneeded: [Sword_2, Sword_3]
Successfully acquired Priest

Unfortunately this shows another problem, I wasn't removing the sword dice from the pool when it matched exactly. Also I wasn't allowing two Sword 3s to fulfil a Sword 4 requirement. A quick fix here which gets us back around to an eighty-one percent chance to acquire the Priest.

Here's a failed acquisition attempt which looks correct:

Reqs: [Sword_4, Sword_1, Political, Castle]
Rolls: [Wizard, Sword_1, Sword_1, Sword_1, Sword_2, Political, Wizard]
Matched: Political
Sword_1 Matched exactly
No more matches with: [Wizard, Sword_1, Sword_1, Sword_2, Wizard], still need [Sword_4, Castle]
Rerolling 5
Reqs: [Sword_4, Castle]
Rolls: [Sword_1, Wizard, Sword_1, Castle, Wizard]
Current Matches: [Political, Sword_1]
Matched: Castle
No more matches with: [Sword_1, Wizard, Sword_1, Wizard], still need [Sword_4]
Rerolling 4
Reqs: [Sword_4]
Rolls: [Sword_1, Political, Sword_1, Political]
Current Matches: [Political, Sword_1, Castle]
No more matches with: [Sword_1, Political, Sword_1, Political], still need [Sword_4]
Rerolling 3
Reqs: [Sword_4]
Rolls: [Political, Political, Sword_3]
Current Matches: [Political, Sword_1, Castle]
No more matches with: [Political, Political, Sword_3], still need [Sword_4]
Rerolling 2
Reqs: [Sword_4]
Rolls: [Sword_3, Castle]
Current Matches: [Political, Sword_1, Castle]
No more matches with: [Sword_3, Castle], still need [Sword_4]
Rerolling 1
Reqs: [Sword_4]
Rolls: [Sword_3]
Current Matches: [Political, Sword_1, Castle]
No more matches with: [Sword_3], still need [Sword_4]
Rerolling 0
Failed to acquire Priest remaining:

Although it does show that perhaps I should have included the 2,1,1 match for Sword 4.

Last Edit: Okay I've made some modifications and now the code supports arbitrary Sword requirements up to twelve. It was a little finicky to get working correctly and my generic sword matching algorithm isn't quite as good as the manual implementation - 36.5% compared to 40% for 3 Sword_4 requirements. One failure of my generic algorithm is that it will match four 1 Swords to a 4 Swords requirement, even if there are additional Sword requirements. I doubt that's the entire cause of the discrepancy.

In any case the code is available here.

The chance to acquire a card with a requirement of 12 Swords is about 11.5% when rolling seven dice.

Answer 3

I reccommend using the website AnyDice to simulate outcomes. You'll need to read the documentation a bit to figure out what to do, but this will be the best way to effectively simulate what you're looking for. Pay special attention to the section on Arbitrary Dice, which will allow you to create the specific dice you're looking for, though keep in mind you're effectively using d6 at all times in this example.

The articles section also has some great information about different programs for in-depth analysis that may be useful to you.