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What Are The Odds?

posted Monday, October 16th 2006 by Jonathan Drain
None of the AboveThird Edition

Back in my entry of 9th of October, you may recall I said that “I once did a statistical analysis of your chance of rolling at least one 18; I’ll have to see if I can find it.” As it turns out I couldn’t find it—so I redid it from scratch using perl. The following information assumes a standard “4d6, drop lowest” roll, with no special extras like rerolls or point shift.

The average roll on 4d6 drop lowest is 12.244, compared to 10.5 on a straight 3d6. Your most likely rolls on a straight 3d6 are 10 or 11; this is the basis of the human average as well as minimum and maximums. Of course, a straight 3d6 rarely generates you a lot of high ability scores.

The odds of rolling an eighteen in one roll are 1.620%, or just over one in sixty-two. However, a character has six rolls arranged as desired – the chance of rolling at least one eighteen are 9.337%, a little better than one in eleven. With straight 3d6, such as rolling up a commoner, your chances of an eighteen would be a mere 0.463% – that’s only one in 216.

So if less than one in ten characters will naturally begin with an eighteen, what can be done to increase your chances of getting one one? One solution is to play a character other than a human for once. A quick calculation reveals that counting in the +2 racial bonus, the chances of getting at least one eighteen or higher are 56.757% – more than half.

Here are the statistics generated by my perl script, for your perusal.

Rolling every possible combination of 4d6 drop lowest...
Made 1296 rolls of 4d6 drop lowest. The average roll is 12.2445987654321.
 
Score   Frequency   Percentage   One or more
-----   ---------   ----------   -----------
3               1       0.077%        0.462%
4               4       0.309%        1.838%
5              10       0.772%        4.541%
6              21       1.620%        9.337%
7              38       2.932%       16.352%
8              62       4.784%       25.482%
9              91       7.022%       35.391%
10            122       9.414%       44.744%
11            148      11.420%       51.692%
12            167      12.886%       56.295%
13            172      13.272%       57.443%
14            160      12.346%       54.643%
15            131      10.108%       47.237%
16             94       7.253%       36.350%
17             54       4.167%       22.536%
18             21       1.620%        9.337%

Comments

  1. What Are The Odds When Rolling 3d6? « Jonathan Drain’s D20 Source: Dungeons & Dragons Blog

    May 25th, 2010

    [...] what are the odds of rolling each ability score on 3d6? I wrote up a quick python script like my 4d6 drop lowest calculator from [...]

  2. DnD Guy

    May 25th, 2010

    Statistics Got to love it

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