This is a probabilistic analysis of Phantom Roll, some possible strategies and the outcome probability distributions for each. It is long; while a full understanding of the methods involved will require some knowledge of math, reading the conclusions does not.

Phantom Rolls can be divided into four groups based on their lucky and unlucky numbers (thanks to Spider-Dan for the complete list). Strategies and their probabilities are a little different for each group. In each group I identified several different strategies which I characterize as "safe", "moderate", and "risky" based on their bust risk. Strategies that reroll more often have a higher bust risk, but also have higher average numerical outcome and a higher chance of hitting 10 and 11. (The chance of hitting the lucky number does not change for any strategy examined in this post since there is no reason not to keep rolling as long as you are below the lucky number, and no good reason to roll if you are *on* it. The lucky number chances I calculated coincide exactly with Icemage's calculations in another thread.)

Methodology (technical): The game was modelled as a digraph in which vertex (n,k) represents the probability of having a total of n after precisely k rolls. Based on the strategy being used (formally defined as a vector of 1's indicating numbers that will be rerolled, and 0's indicating numbers that will be kept), some vertices are terminal while others have outgoing connections to (n+1,k+1) through (n+6,k+1). In order to shorten the calculations, it was assumed that a maximum of 6 rolls could be made. Calculations were performed by an OpenOffice 1.1 spreadsheet using formulas of the form (innerproduct( [p(n-6,k-1),p(n-1,k-1)] , [reroll(n-6),reroll(n-1)] ) / 6 (using dummy entries with 0 probability when necessary), and then summing across rows for terminal results. All strategies have 99+% probability of proceeding to completion (either a number you want to keep in that strategy, or a bust) within 6 rolls.

Group 1 - 2/6 Rolls (Magus, Choral, Samurai)

Safe strategy: Reroll any number 6 or less, except 2 (which is lucky).

Lucky 2: 19.44%, 7: 19.36%, 8: 16.58%, 9: 16.58%, 10: 13.34%, 11: 9.56%, Bust: 5.15%

Moderate strategy: Reroll any number 7 or less, except 2.

Lucky 2: 19.44%, 8: 19.80%, 9: 19.80%, 10: 16.56%, 11: 12.78%, Bust: 11.60%

Risky strategy: Reroll any number 8 or less, except 2.

Lucky 2: 19.44%, 9: 23.10%, 10: 19.86%, 11: 16.08%, Bust: 21.49%

Group 2 - 3/7 Rolls (Healer's, Monk's, Drachen, Gallant's)

Safe strategy: Reroll any number 5 or less, except 3 (which is lucky); also reroll 7, which is unlucky. Keep 6.

Lucky 3: 22.69%, 6: 30.88%, 8: 13.80%, 9: 10.56%, 10: 10.56%, 11: 6.78%, Bust: 4.74%

Moderate strategy: Reroll any number 7 or less, except 3.

Lucky 3: 22.69%, 8: 19.80%, 9: 16.56%, 10: 16.56%, 11: 12.78%, Bust: 11.60%

Risky strategy: Reroll any number 8 or less, except 3.

Lucky 3: 22.69%, 9: 19.86%, 10: 19.86%, 11: 16.08%, Bust: 21.49%

Group 3 - 4/8 Rolls (Ninja, Hunter's, Chaos, Puppet, Beast, Warlock's)

Safe strategy: Reroll any number 5 or less, except 4 (which is lucky); also reroll 8, which is unlucky.

Lucky 4: 26.47%, 6: 30.88%, 7: 14.21%, 9: 10.10%, 10: 6.32%, 11: 6.32%, Bust: 5.72%

Moderate strategy: Reroll any number 6 or less, except 4; also reroll Unlucky 8.

Lucky 4: 26.47%, 7: 19.36%, 9: 16.10%, 10: 12.32%, 11: 12.32%, Bust: 13.43%

Risky strategy: Reroll any number 8 or less, except 4.

Lucky 4: 26.47%, 9: 19.86%, 10: 16.08%, 11: 16.08%, Bust: 21.49%

Group 4 - 5/9 Rolls (Corsair's, Evoker's, Rogue's, Fighter's, Wizard's)

Safe strategy: Reroll any number 4 or less, and Unlucky 9.

Why is this "safe" when you're rerolling a 9? Because a bust isn't much worse than an unlucky number (compared to turning it into a 10 or 11, which is a major improvement), and your chances of getting *either* of the bad outcomes is quite low due to the overall small amount of rerolling.

Lucky 5: 30.88%, 6: 30.88%, 7: 14.21%, 8: 11.43%, 10: 5.78%, 11: 1.37%, Bust: 5.46%

Alternative safe strategy: Reroll any number 6 or less, except for 5; if you get stuck on Unlucky 9, keep it as preferable to a 2/3 chance of busting.

Lucky 5: 30.88%, 7: 19.36%, 8: 16.58%, Unlucky 9: 13.34%, 10: 9.56%, 11: 5.15%, Bust: 5.15%

As you can see the bust chance doesn't change much, but there is now a significant unlucky chance too. However the 10-11 chances are slightly improved.

Ultrasafe strategy: Only reroll 1 through 4.

Lucky 5: 30.88%, 6: 30.88%, 7: 14.21%, 8: 11.43%, Unlucky 9: 8.19%, 10: 4.41%.

Bust is impossible under this strategy, the worst that can happen is getting stuck with unlucky 9; however, 11 is also impossible and the chance of a 10 is under 5%. It pretty much amounts to giving up, and probably getting a weak result, if you don't get your 5.

Moderate strategy: Reroll any number 6 or less, except for 5; also reroll Unlucky 9.

Lucky 5: 30.88%, 7: 19.36%, 8: 16.58%, 10: 11.78%, 11: 7.37%, Bust: 14.04%

Risky strategy: Reroll any number 7 or less, except for 5; also reroll Unlucky 9.

Lucky 5: 30.88%, 8: 19.80%, 10: 15.54%, 11: 11.13%, Bust: 22.63%

It's up to you which strategy to choose for any given roll and situation; there isn't necessarily any one right answer to how much risk to take. But it's better to do so with knowledge of the potential risks and benefits of each course of action.