Monday, January 26, 2015

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.


  1. Not necessarily entirely related, but this reminded me of a player at our FLGS who was pretty notorious for fudging the rules to benefit himself.

    He played BA and his storm-shield/TH termies all had master-crafted. So whenever he assaulted he would roll all the hits for the squad at once and re-roll any 1s. In reality, master-crafted only allows one re-roll per model so its possibly that 2 of those 1s would have came from the same termi if he had rolled them one model a time like he should have. This could have led to less re-rolls than he gave himself.

    I'm not sure that this was necessarily malicious, but there are a lot of people who try to simplify mathematical situations like this or the one you presented without realizing that subtle differences actually change the odds.

    1. Any missed hit* not just 1s. Sorry.

    2. For the same reason you always want to roll plasma guns separately. Did one guy get hot twice, or did two guys get hot once?

  2. Always enjoy some math hammer. Good to know the basic odds...similar to playing blackjack...should I hit? Or stay? :)

  3. Yep. :-) My easy thought was:
    If I reroll a "3" I have 3 better result, 2 worst result and one equal. So it's better to reroll "3s". Rerolling a "4" I will have only 2 better result. 3 worse, and one equal.

  4. I have a very difficult question for you!

    How much dangerous deep striking is (for the deep striker)? I mean, there are a lot of variables: hit or scatter, dice direction, distance direction of the scatter... And all I wanna know is: where should I deepstrike without risking a disaster? (I mean, a delay is not a disaster as "unit destroyed"....)

    PS: I know that is our reader's fault, but... aren't you afraid that your blog is becoming " The War-math-er of fun"? ;-)

  5. This is a question I have been considering for some time, and the geometry can get complicated, but I will see what I can do.

    I love math. Yesterday I spent most of my free time working on one problem. If other people find it half as enjoyable (or at least as useful) as I do then I am certainly not worried.

  6. Some days ago I have seen a graphic but without explanation. It could be a function whit distance from enemy model and % of risk but wasn't explained so I am not sure it was about that. But maybe you can find if something similar is already been done on the net (and understand it better than I can do!)

    I find Math really interesting and funny, but I have no time to dedicate at it! So I am happy that you write on this blog.... I always read it and I appreciate all the "dirty work" you do for us! :-D
    (last year I usually read a math blog written by an Italian math professor... sadly he do not write anymore... I guess interesting general math problems have an end...)