Re-roll vs. Roll Two Pick the Highest

There are a couple of situations that many people think are exactly the same (mathematically), but are actually quite different.  Most of these situations derive from rolling multiple dice and picking, presumably the highest result, vs rolling once with the option to re-roll the result.  Thinking back to my first post about math I am going to use expected value to try and maximize the various situations.

Roll 2 Picking the Highest:

Lets look a situation like ordinance first; it is the simpler of the two.  In that situation we have these probabilities.

1: 1/36
2: 3/36
3: 5/36
4: 7/36
5: 9/36
6: 11/36

With an average of 5.25

It is pretty straightforward since there isn't any re-rolling.

Most times we encounter re-rolling we are dealing with a pass/fail situation such as re-rolling your chances to hit.  In a pass/fail situation re-rolling a miss and rolling two dice while picking the highest are exactly the same.

This changes once our results can be gradient.  The two situations I can think of (there may be more) are re-rolling your run distance with fleet, and re-rolling your armor penetration (if you get a glance) with tank hunter.  Both of these are special situations, because if you re-roll there is the possibility to roll worse than you did the first time.  The key to maximizing your success is to  know when to re-roll and when to keep your result.

There are sometimes when both fleet and tank hunter can be described as pass/fail.  If you are running to capture an objective you are either within range or out of range.  If you are trying to knock the last hull point off the vehicle, you don't necessarily care about getting a penetrating result.

Re-Rolling Your Run Result:

If you are trying to maximize your run distance you can choose if you are going to keep your result, or if you want, to re-roll.  Rolling a 1 means that you are always going to want to re-roll, because every result is equal to or greater than the original result.  Similarly you would never re-roll a 6; however, where is the cutoff?  Is there a point where it is statistically better to always re-roll? Never re-roll?

Without any re-rolling, a single dice roll will have an expected average of 3.5 and every result will have the same probability of 1/6.

Re-Rolling 1's:
If I just re-roll my 1's and keep the other results my probabilities switch to
1: 1/36
2: 7/36
3: 7/36
4: 7/36
5: 7/36
6: 7/36

With these new results the expected value becomes 3.92, so just re-rolling ones is already increasing out expected average.

Re-rolling 1's and 2's:
Now let's see what happens if I re-roll both 1's and 2's

1: 2/36
2: 2/36
3: 8/36
4: 8/36
5: 8/36
6: 8/36

And the expected average becomes 4.17

Re-Rolling 1's, 2's, and 3's

1: 3/36
2: 3/36
3: 3/36
4: 9/36
5: 9/36
6: 9/36

With an average of 4.25

Re-rolling 1-4

1: 4/36
2: 4/36
3: 4/36
4: 4/36
5: 10/36
6: 10/36

With an average of 4.17

Re-Rolling everything but 6's

1: 5/36
2: 5/36
3: 5/36
4: 5/36
5: 5/36
6: 11/36

With an average of 3.92

To summarize you can maximize your overall result when you re-roll 1, 2, and 3, so with a run move re-rolling 1-3 essentially gives you an average extra 0.75 inches.

If we apply this result to the re-roll for tank hunter here is how we can interpret things.  If you roll a glance on a 1, 2, or a 3 then you should re-roll your result and try for the penetrating hit.  However, if you glance on a 4, 5, or 6 then keep the result.

Now you know.