Shannon Appelcline
United States Berkeley California

When I was writing up my review (see the Links section) I was curious what the probabilities were of rolling each of the prizes naturally on 4 dice.
Here's my best work at the calculations, in order from easiest to hardest:
1. Exactly 7, Exactly 13: 96/1296 =~ 7.4% 2. Straight: 1/18 =~ 5.6% 3. 3 or Less, 17 or More: 35/1296 =~ 2.7% 4. 3 of a Kind (odd or even): 1/72 =~ 1.4% 5. Two Pair: 15/1296 =~ 1.2% 6. 4 of a Kind: 1/216 =~ 0.5%
What's less clear to me are if the actual chances of rolling any prize, including the rerolls, is notably different. This *isn't* just a question of probability, but also psychology.
For example, I list the straight as one of the most common prizes, and I'm entirely sure of the math in that one, but in actuality it only came up once in our game, and no one took the prize from the original owner. I suspect this was a result of the psychology of the players, where they were less likely to pursue the straight because it "seemed harder".
There's also some question of how overlapping probabilities might change up what people are willing to go for: if a player can reroll one die and go for two different, less likely prizes; or roll a different die and go for one more likely prize; then he'll probably go for the two less likely ones, as long as the sum of their probabilities is higher than the singular probability. Again, I don't know how this interactivity might influence final odds.
Nonetheless, despite my disclaimers, the above list is probably a good thumbnail for which prizes will be hardest to take from you, namely the 4 and 3 of a kind, and which will be easiest to take, namely the exactly 7 and 13.

Shannon Appelcline
United States Berkeley California

The Math
Here's my math for those calculations:
Exactly 7, Exactly 13: This is based on figuring out how many ways the number can be put together from the component die. I'm sure there's some simple formula to figure it out, but I don't know it, and it's not easy to calculate because it's a bell curve, which means that its nice progression at the edges goes wonky in the middle.
By my best method of figuring out all the possibilities, I came up with 96 different ways to form each of 7 and 13. There are 1296 possible options on four sixsided dice (6^4);hence 96/1296.
Straight: This one's easy. There are 3 potential straights (starting with 1, 2, and 3), and since there are each four dice, there are 4! potential ways to arrange each straight.
Hence: (3 * 4!) / 1296 = 72/1296 = 1/18
3 or Less, 17 or More: Again, an exercise in counting the possibilities. Here I had to add the ways to form "0", then "1", then "2", then "3". Fortunately, this is at the bottom of the bell curve, and the numbers are nice & consistent. 17 is exactly the same as 3, statistically.
(1 + 4 + 10 + 20) / 1296 = 35/1296
3 of a Kind, Odd or Even: This is a standard probability. For each of them (odd or even), the first die just has to be an odd or even (1/2), then you have to have another die that is the same eveness, but doesn't match (1/3), and two others that do (1/6). There are 3 different ways to put those last three dice together. This is one of the ones that I'm not totally confident on, but I think it's right:
3 * (1/2 * 1/6 * 1/6 * 1/3) = 1/72
Two Pair: Similar deal: the first die can be anything, and then you need one die that's different, and then one that matches each of the ones before, and again there are three ways to organize the latter three dice. Again, not 100% sure, but it looks good to me:
3 * (1 * 5/6 * 1/6 * 1/6) = 15/1296
4 of a Kind: And this one's simple: one die that can be anything, and three that match it:
1 * 1/6 * 1/6 * 1/6 = 1/216

Shannon Appelcline
United States Berkeley California

One Correction
After carefully calculating all those probabilities, I couldn't get the math right!
If you like at my two pair, you'll see I list it as 3 * (1 * 5/6 * 1/6 * 1/6). That's 15/216, not 15/1296, or 6.9%, which makes it the second most likely result, as you'd expect.
Whew.
1. Exactly 7, Exactly 13: 96/1296 =~ 7.4% 2. Two Pair: 15/1296 =~ 6.9% 3. Straight: 1/18 =~ 5.6% 4. 3 or Less, 17 or More: 35/1296 =~ 2.7% 5. 3 of a Kind (odd or even): 1/72 =~ 1.4% 6. 4 of a Kind: 1/216 =~ 0.5%
So, hard to steal: 4 or 3 of a kind; easy to steal: exactly 7 or 13 or two pair, possibly straight depending on psychological issues.

Andrew Juell
United States Buford Georgia

Re:One Correction
Shannona (#79152),
By direct enumeration, I found 104/1296 combinations yielding a sum of exactly 7, and similarly with exactly 13).
0025 x12 0034 x12 0115 x12 0124 x24 0133 x12 0223 x12 1114 x 4 1123 x12 1222 x 4
Also, the singleparity threeofakinds are a bit more likely than your calculation suggests: there are 24 ways of making each of them, for odds of 1/54 rather than 1/72.
There are 4 ways of selecting which of the dice will not be a member of the triplet, 3 ways of selecting the value of the dice in the triplet, and 2 remaining values for the singlet. 4*3*2=24.
Also, for purposes of actual play, the chance of getting one of these 'naturally' doesn't necessarily imply a corresponding difficulty of obtaining them with the four available rolls.


