Heavy duty mathhammer as requested in another post.
Posted in a fresh thread for location by board searches.
You have been warned.
Rerolling improvement over not rerolling varies with the target number (note that this applies to all rolls vs. target numbers except for enemy saves which run conversely and will be address in a second table).
Success Chance with and without reroll vs target number of 1d6 rolls (all except enemy armor saves). roll/target#  2+  3+  4+  5+  6+  no reroll  83.34%  66.67%  50.00%  33.33%  16.67%  with reroll  97.22%  88.88%  75.00%  55.55%  30.55%  improvement  13.88%  22.22%  25.00%  22.22%  13.88%  % improvement  16.66%  33.33%  50.00%  66.67%  83.55%  First row is the chance of successfully getting the result without reroll. Second row is the chance of successfully getting the result with a reroll. Third row is the improvement with a reroll (base chance + improvement = rerolled chance).* Fourth row is the percentage improvement (rerolled chance / base chance).**
Note: *Due to the nature of calculating the improvement (failure chance x success chance = improvement) the largest improvement occurs where the chance of failure AND success are both greatest (4+), not at the extremes (6+ or 2+). **Due to adding base chance to the improvement to determine your reroll chance (base chance + improvement = rerolled chance) the percentage improvement goes up as your base chance of success goes down.
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Since a successful enemy armor save is a failure for you the odds essentially flip, here is the table for them (note that the save targets are essentially transposed).
Success Chance with and without reroll vs target number of 1d6 rolls (enemy armor saves). roll/target#  6+  5+  4+  3+  2+  no reroll  83.34%  66.67%  50.00%  33.33%  16.67%  with reroll  97.22%  88.88%  75.00%  55.55%  30.55%  improvement  13.88%  22.22%  25.00%  22.22%  13.88%  % improvement  16.66%  33.33%  50.00%  66.67%  83.55%  First row is the chance of successfully getting the result without reroll. Second row is the chance of successfully getting the result with a reroll. Third row is the improvement with a reroll (base chance + improvement = rerolled chance).* Fourth row is the percentage improvement (rerolled chance / base chance).** Note:
*Due to the nature of calculating the improvement (failure chance x success chance = improvement) the largest improvement occurs where the chance of failure AND success are both greatest (4+), not at the extremes (6+ or 2+).
**Due to adding the base chance to the improvement to determine your reroll chance (base chance + improvement = rerolled chance) the percentage improvement goes up as your base chance of success goes down.
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In regards to bladestorm or any ability the occurs on a single occurrence (Distort, Monofilament, etc.) of a 6, if you have a reroll, the higher the target number (more of a chance to fail, the greater the chance to get the trigger result of a 6.
Successfull Special Ability Activation vs. a target number (Bladestorm, Distort, Monofiliment, etc.). roll/target#  2+  3+  4+  5+  6+  no reroll  16.67%  16.67%  16.67%  16.67%  16.67%  with reroll  19.44%  22.22%  25.00%  27.78%  30.56%  improvement  02.78%  05.56%  08.33%  11.11%  13.89%  % improvement  16.62%  33.29%  50.00%  66.65%  83.32% 
First row is the chance of successfully getting the activation result (roll a 6) without reroll. Second row is the chance of successfully getting the result with a reroll. Third row is the improvement with a reroll (base chance + improvement = rerolled chance).* Fourth row is the percentage improvement (rerolled chance / base chance).**
*Due to the nature of calculating the improvement (failure chance x success chance = improvement) and since the chance of success is fixed, the largest improvement occurs where the chance of failure is greatest (6+).
**Due to adding the fixed chance of success to the improvement to determine your reroll chance (base chance + improvement = rerolled chance) the percentage improvement goes up as your base chance of success goes down.
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In regards to Spirit mark or an ability that allows rerolls on a 1 (Bolter Drill, etc.) the lower the target number (more of a chance to succeed, the greater the amount of improvement.
Successfull chance vs. a target number (Spirit mark, Bolter drill, etc.). roll/target#  2+  3+  4+  5+  6+  no reroll  83.33%  66.67%  50.00%  [td33.33%  16.67%  with reroll  97.22%  77.78%  58.33%  38.89%  19.44%  improvement  13.89%  11.11%  08.33%  05.56%  02.78%  % improvement  16.67%  16.67%  16.67%  16.67%  16.67% 
First row is the chance of success without reroll. Second row is the chance of success with a reroll triggered by rolling a 1. Third row is the improvement with a reroll triggered by rolling a 1 (base chance + improvement = rerolled chance).* Fourth row is the percentage improvement (rerolled chance / base chance).**
*Due to the nature of calculating the improvement (trigger chance x success chance = improvement) and since the trigger chance is fixed, the largest improvement occurs where the chance of success is greatest (2+).
**Due to adding the fixed chance of success to the improvement to determine your reroll chance (base chance + improvement = rerolled chance) the percentage improvement remains the same throughout the table.
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The chance to penetrate with rerolls (Doom, Tank Hunter, etc.) varies with the required target number and your decision to either keep or reroll glances. If you reroll glances, your chance at total failure increases. To read the chart below subtract the Strength of the hit from the AV of the armor, which in most cases will give you a resulting target number. If your result is 0 all hits automatically penetrate (S10 vs AV10). If your result is 1 all hits will score at least glancing hits (use 1+). If your result is between 2 and 6 consult the appropriate row. If it is greater than 6, but your weapon has a special ability that activates on a 6 (Distort for example) use 6+.
Successfull Penetration vs. AV (1014). target#/rolltype   No RR    RR Fails    RR Glances   target#/result  Fail  Glance  Success  Fail  Glance  Success  Fail  Glance  Success  1+    16.67%  83.33%          02.78%  97.22%  2+  16.67%  16.67%  66.66%  02.78%  19.45%  77.78%  05.56%  05.56%  88.88%  3+  33.33%  16.67%  50.00%  11.11%  22.22%  66.66%  16.67%  08.33%  75.00%  4+  50.00%  16.67%  33.33%  25.00%  25.00%  50.00%  33.33%  11.11%  55.55%  5+  66.66%  16.67%  16.67%  44.44%  27.78%  27.78%  55.55%  13.89%  30.56%  6+  83.33%  16.67%    69.45%  30.56%         




Modified to add a table to cover abilities that trigger rerolls on a 1.