Basic Odds of Shadowfist by Mike Nickoloff
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[posted 11 Aug 2006]
This article was originally posted on Bryant Durrell's Innocence site Nov.4, 1998, then hosted for a long time at Jon Challis' site kelanen.fsnet.co.uk/shadowfist after Bryant's site closed down, but Jon's site disappeared in early 2004, so I've mirrored the article here.
This article is being presented to show some misconceptions about the optimum number of feng shui sites and foundation characters in a deck.
The calculations below were determined using Excel and the iteration routine was programmed using Java. The values are meant to be guidelines; they should be only interpreted with 2 digits of accurracy. The pseudo-random generator should be used as an approximation.
12 6 60 75
15 6 60 84
18 6 60 90
12 6 50 83
15 6 50 90
18 6 50 94
12 8 60 85
15 8 60 92
18 8 60 95
12 8 50 91
15 8 50 96
18 8 50 98
For real life playing, assume 8 cards is discarding 2 non-good cards after drawing your initial 6. This approximation is good enough for our purposes.
What does this mean? If you play with 1/5 feng shui sites in a 60 card deck, you only have a 75% chance of drawing one in your opening hand. For a 4 round tournament, you should get screwed once. Is this acceptable? Probably not. You cannot use that table to figure out the chances of drawing a feng shui site and a foundation character.
You must use hypergeometric distributions to find the answers. With a decent math background, you can easily understand the math behind the calculations. However, our problem stems from the quantity of valid hands that can be produced. I did some preliminary calculations using this method, and they seemed low. Therefore, I stopped and concluded my calculations using a Monte Carlo method. I wrote a Java program to simulate drawing a hand. A valid hand was one that produced both a feng shui site and a foundation character. I assumed the # of each were the same with respect to each other. Also, draws were not-replaced, just like a real deck. Then, I iterated a number of hands which produced a approximate percentage; well enough for us.
D N %
Just to make sure
I was on the right track, I repeated C=12, N=60, D=6 for 50k and 100k iterations:
Accurate to 2 decimal places; looks fine.
What can we conclude? First, drawing 8 cards increases our chances around 10-15% than if we only draw 6. The numbers would increase more if you discard even more cards. Therefore, discarding heavily (even in a 50 card deck) should be a priority. Playing with 4 cards toasted in the early game is much better than stalling out.
A smaller deck with the same number of FS/FC cards (but increased proportions) decreases the chance that you will stall. Therefore, you could play a huge deck, but allowing to discard heavily (seeing at least 10 cards); however, you may then toast cards that you have very few in the deck.
What is optimal? 100% would be nice, but this is impossible. As basic laws of probability state that each trial is independent from each other, a statement like, "If I play 5 rounds, and I have a 80% chance of stalling, I should only stall once.", is misleading. The average chance to stall is 20%, but you may stall twice, or all 5 rounds. (That calculation is for another article entirely)
Each person is able to make their own decision on deckbuilding. It is these choices that effect our gameplay the most. Wouldn't you give yourself the best advantages possible?
Mike Nickoloff, Nov.4, 1998