## A Feast for Probabilities

Tuesday, 22nd February 2022 ◆ Looking for French one in strange night (7) Maths BGAIn somewhat of a break from Uwe Rosenberg's usual worker placement games, A Feast for Odin introduced an element of luck in the form of die rolls. These die rolls come in two flavours:

- Hunting/snaring/whaling - where you aim to roll low
- Raiding/pillaging - where you aim to roll high

Under normal circumstances, you are allowed up to three rolls to get the result you want. After you stop rolling, you may spend resources and cards to further modify the result. I never really know how many resources and cards I need before I should risk using these spaces, so I thought I'd do a quick probability run-down. This also serves to determine how large the element of luck actually is...

I'm not going to look at raiding and pillaging here because it is a more complex case. Instead of a simple success/failure, you can pick different rewards depending on how high the result is.

### Hunting, snaring and whaling

With hunting, snaring and whaling, you are aiming to get a result of 0. Your base value is the value of the die after you finish rolling, and you have up to three rolls in total. Then, you subtract 1 for each card or wood that you spend. In the case of whaling, you can also subtract more depending on the amount of ore you have on whaling boats. Furthermore, whaling a d12 whilst hunting and snaring uses a d8.

If you have \(x\) resources at your disposal, and if \(X_k\) is the random variable representing a roll of a die with \(k\) faces, the chance of success can be calculated as:

$$ \begin{aligned} \mathbb{P}(success) &= 1 - \mathbb{P}(failure) \\ &= 1 - \mathbb{P}(X_k > x)^3 \\ &= 1 - (\frac{k - n}{k})^3 \end{aligned} $$

Evaluating thing and creating a graph:

What is surprising, is that the probability of success is generally quite high. If you have 2 snare cards, you already have a 58% chance of success (and if you fail, you are refunded a viking and some resources, so the downside is not that severe).

In short, I think the luck aspect of AFFO is much more minor that it appears at first blush. And, I will try and use these spaces a lot more going forward!

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