# 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?”

Thanks guys!

• This might be easier to do on paper with some very basic combinatorics rather than by programming anydice to handle it for you. May 28 at 21:35

### 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?