xG (Expected goals) is a beloved statistic in football (European..). In a dice game it’s even easier to calculate and compare.
To make it simple, I have one position, and it is only having one resource: a 2 grain.
My xCard for a throw is 1/36, as I statistically will earn one card per 36 rounds. Then we build this up, and I have:
2grain (1/36)
4wool (3/36)
6stone (5/36)
8stone (5/36)
10wood (3/36)
11grain (2/26)
My xC for a throw is then 19/36 = 0.53. It goes a round (4 players) without gaining any cards, and my xC is now 4*0.53 = 2.12.
Then someone places the robber at my 8 stone, and throws the dice. My xC at that throw is reduced, as I cannot get cards from 8 stone. Let’s say the dice shows 6. My score is now 1 gained card, vs an xC of 2.12 + 14/36 = 2.51.
–
Moving on to xD (Expected Discards), it will follow the same pattern: I sit with 8 cards on hand, and the next player is to roll the dice. The probability of rolling 7 is 16.7%, so my expected loss is 4 cards * 16,7% = 0.67 cards. Instead I have a very lucky dice, and get 3 more cards. I have now 11 cards, and the dice is to be rolled. I now have a 16,7% chance of losing 5 cards, so my xD increases with another 0.83.
–
At the end of the game I can now see that I’ve earned 73 cards through the dice, but my xC is actually 76.9. I had bad luck! On the other hand, I only lost 4 cards by sevens, while my xD in the game in total was 8.4. So lucky me, the sevens came mostly when I was under the limit.