# Anydice Statistics for the Favored by Fortune system

Quick question about how to do the following in Anydice. I’m trying to figure out what the statistics are for a game called Household currently on Kickstarter that utilizes the Favored by Fortune system by 2LM.

The game uses d6 with custom sides. There are 4 special unique suits, then a blank, and a Joker side.

Success is when two or more dice show the same side. Jokers are wild and can count as any side. Blanks count as nothing.

I assume you can roll anywhere from 2-5 dice.

What I’m trying to determine is what are the chances of getting duplicates, triplets, etc on these dice. Success vs Failure. Then I’d probably be interested in “what are the chances of getting successes of specific sides?”

• This might be easier to do on paper with some very basic combinatorics rather than by programming anydice to handle it for you. Commented May 28, 2022 at 21:35
• It looks like in the latest (December 2022) Quickstart they went back to something closer to the Broken Compass system. Commented Dec 29, 2022 at 1:43

### The quickstart guide has a table...

2LM has provided a quickstart guide (ZIP) on the official Household website. Page 60 has a table of probabilities:

DICE BASIC (pair) CRITICAL (triple) EXTREME (quad) IMPOSS (quint+)
2 36% - - -
3 58% 12% - -
4 75% 29% 4% -
5 83% 47% 13% 1%
6 91% 61% 26% 5%
7 95% 73% 42% 14%
8 98% 81% 56% 26%
9 99% 87% 68% 40%

### ...but the probabilities are not correct.

For example, consider the possibilities for rolling a triple on 3 dice:

• 3× Joker: 1 way
• 2× Joker, 1× of one suit: 4 suits × 3 unique orderings = 12 ways
• 1× Joker, 2× of one suit: 4 suits × 3 unique orderings = 12 ways
• 3× of one suit: 4 suits = 4 ways

Total: 29 ways out of 216, or about 13.43%, versus the table's 12%.

Or, consider the possibilities for not rolling anything on 3 dice:

• 3× blank: 1 way
• 2× blank, 1× of one suit: 4 suits × 3 unique orderings = 12 ways
• 2× blank, 1× Joker: 3 unique orderings = 3 ways
• 1× blank, 2× of different suits: 6 combinations of 2 suits × 6 unique orderings = 36 ways
• 3× of different suits: 4 combinations of 3 suits × 6 unique orderings = 24 ways

Total: 76 ways out of 216, or about 35.19% chance of no success. That's considerably less than the 42% implied by the table.

One could try to interpret the clause

a Joker can be combined with any other Ace

as not counting pure-Joker rolls, but this isn't enough to repair the probabilities, and it also causes the 2-pool probabilities to diverge from the table.

### Computed probabilities

Here are probabilities computed using my Icepool Python package:

DICE BASIC (pair) CRITICAL (triple) EXTREME (quad) IMPOSS. (quint+)
2 36.11% 0.00% 0.00% 0.00%
3 64.81% 13.43% 0.00% 0.00%
4 83.56% 34.03% 4.71% 0.00%
5 93.49% 55.56% 15.56% 1.61%
6 97.75% 73.02% 31.14% 6.55%
7 99.30% 84.95% 48.13% 15.56%
8 99.80% 92.23% 63.63% 27.84%
9 99.95% 96.31% 76.18% 41.76%

You can run the script in your browser using this JupyterLite notebook.

### Some other ambiguities in the quickstart

• What happens in the unlikely case that the player rolls more than 5 of a kind? Does it have any extra effect? If not, can e.g. a set of 7 be split into a quint and a pair?
• On p. 54, the rules state that "You may play an Ace to [...] Re-roll all dice that aren't part of a combination after your initial roll." If the player rolls one Joker and the rest blank on their initial roll, are they obligated to re-roll the Joker? Conversely, if they roll two pairs on the initial roll, may they re-roll one of the pairs?

## December 2022: newer quickstart, older rules

In December 2022, 2LM released a new version of the Household quickstart on DriveThruRPG. It seems they elected to go back to the older version of the system as used in their previous title, Broken Compass. The basic mechanic works as follows:

1. Roll a pool of between two and nine d6s.
2. Count matching sets.
• 2-of-a-kind: Basic success (valued at 1 point).
• 3-of-a-kind: Critical success (valued at 3 points).
• 4-of-a-kind: Extreme success (valued at 9 points).
• 5-of-a-kind: Impossible success (valued at 27 points).
• 6-of-a-kind: What a Hero! (valued at 81 points). (As of December 2022 this is only listed in Broken Compass; it remains to be seen whether Household adopts this in the final version.)
3. Total up the points. (For purposes of plotting only. The point values reflect the "3X1" rule, though by the rules the exchange only goes one way.)

The first quickstart had a 2:1 ratio between success levels; this may reflect the fact that the blanks made it difficult to ensure Basic success while at the same the Jokers made Impossible success more achievable.

## Rerolling

Broken Compass also has a core reroll mechanism.

• The character can potentially reroll up to two times.
• Only numbers that are not part of a success (i.e. only one of that number was rolled) can be rerolled.
• If the character rolled at least one success in the initial roll, they make take the first reroll ("Risk").
• If the first reroll does not add or improve any successes, they lose one of their existing successes (of their choice) and the roll is over.
• If the first reroll adds or improves at least one success, the character may either accept the result or make a second reroll ("All or nothing"). (The Household quickstart suggests that there will be a similar "All In" mechanic in the final version, but there are no explicit rules yet.)
• If the second reroll does not add or improve any successes, all successes are lost.

We'll analyze two strategies:

1. "Known-difficulty": The difficulty is known; stop when that success level is reached.
2. "Always-reroll": Take every possible reroll, even at the risk of busting.

For known-difficulty, for any decision point, the highest difficulty reached is the max of the current roll (since the player can always choose to exit if the successes are sufficient) and the potential reroll (which the player will surely take otherwise). If the result of the roll is a binary pass/fail, knowing the target difficulty is as good as knowing exactly what numbers will come up on your rerolls!

For always-reroll, we just compute the result of the second reroll (if the reroll is available).

The dotted lines represent the results of always-reroll, which can be thought of as the maximum penalty for taking risks.

Interestingly, there are a few places where extra dice appears to hurt the always-reroll strategy. My guess is that the extra dice can increase the chance of rolling well enough that it would be silly to continue, but always-reroll does so anyways.

## Expertise

If the character has Expertise in the task:

• They can make the first reroll even without successes in the initial roll.
• They don't lose any successes if the first reroll doesn't add or improve any success.
• They can make the second reroll ("All or nothing") even if the first reroll doesn't add or improve any success.

For large pools and low target successes, always-reroll can perform worse than in the non-Expertise case! This is due to the increased possibility of bad bets on "All or nothing".

## Script

You can run the script in this JupyterLite notebook. There are a lot of subtleties in the rerolling process, so let me know if you spot any discrepancies.