You?re running a game in which the
party is ambushed by 45 archers. Arrows
begin flying; you resign yourself to rolling
your 20-sided die over and over (and over
and over and . . .). Anxious players drum
their fingers as they await their turn. Is
there a quicker way? Obviously there is,
or you wouldn't be reading this article. 

Included are 3 tables that will
greatly expedite the die-rolling process.
They emulate, respectively, 5 rolls, 10
rolls, and 20 rolls of a 20-sided die. The
numbers across the top indicate the "to
hit" number (1 is left off since there is
always a 100 percent chance of rolling a 1
or better). The left-hand side shows the
number of hits. The numbers in the table
itself give percentages for successful hits.

To use the tables, determine what the "to
hit" value is. Look up that value on the
column heading of the appropriate table;
this is the column you will be using. Roll
percentile dice. Find the largest value in
the column that is less than or equal to the
number just rolled, and consult the leftmost
number in that row. This represents
the amount of successful hits.
 

Example: In the above scenario, the DM
determines that 10 arrows are being
aimed at Francis the Cleric. The DM consults
the 10-Roll Binomial Table. The
archers need, say, a 12 to hit, and the DM
rolls a 39 on percentile dice. The largest
number less than or equal to 39 in the "12"
column is 27. This corresponds to 4 hits
tread across to the leftmost column) out of
the 10 attempts.

It should be noted that the tables are not
limited to just combat rolls; they can be
used anytime a 20-sided die is rolled to
determine success or failure.

Example: On his turn, Francis decides to
run for help, and casts a sanctuary spell.
The DM must now roll saving throws for
all 45 archers, so he makes two rolls on
the 20-Roll Binomial Table and one roll on
the 5-Roll Binomial Table. Each archer?s
saving throw is 15, so that is the number
of the column referred to on both of the
Binomial Tables. For the first use of the 20.
Roll Table, the DM gets a 77 on percentile
dice. Reading down the ?15? column, he
finds that a result of 77 corresponds to
eight successful saves out of the first 20
attempts. For the next use of the 20-Roll
Table, he gets a 34 on percentile dice.
Again referring to the ?15? column, the
DM finds that 24 is the closest number to
34 that is also lower than 34. Reading back
across that row to the leftmost column, he
sees that only five archers out of the second
group of 20 were able to save. For the
single roll on the 5-Roll Table the DM gets
a 21 on percentile dice, indicating that one
archer out of the final group of five made
his save. Thus, a total of 14 (8 + 5 + 1)
archers are free to shoot at poor Francis
as he flees.

Special cases
Some of the columns have more than
one entry of 99 or 00 in them. Special
procedures must be taken if these numbers
are rolled. Each case is handled differently.
Normally, if the percentile dice
roll is lower than the lowest number in
the column, it indicates no hits. However,
if 00 is rolled (a 00 indicates zero rather
than the usual one hundred) and at least
one 00 entry appears in the column being
referred to, then the DM must add an 
extra number of 20-sided die rolls to the 
number that still remained to be determined 
and continue the process.  The 
number of extra rolls is equal to the number 
of 00 entries in the column being used. 

Example: The sharp-eyed archers who
made their saving throws notice Francis
running away. Each of the 14 archers now
needs a 10 to hit (because Francis? back is
turned). The DM consults the 10-Roll Binomial
Table for the first 10 attempts, and
rolls a 00. There are two entries of 00 in
the ?10? column, which means that all but
two of the first 10 archers have certainly
missed, but there is still a chance that two
of them will hit. The DM adds two 20-
sided die rolls to the four that remained to
be made, for a total of six. Then, using the
5-Roll Binomial Table to handle five of
those rolls, he gets a 01. This is less than
the lowest number in the ?10? column, so
none of those five shots were successful,
For the final archer, the DM rolls a 20-
sided die and gets a result of 7 ? another
miss! Francis manages to escape.

In the case of a 99, the procedure is
similar -- but the number of extra rolls
equals one less than the number of 99's. If
there is only one 99 in the column being
used, then all hits are automatically successful
and no extra rolls are granted.

Example: After Francis escapes, 20
archers are sent after him, and sure
enough they find him (this is just not Francis
's day). Each of them needs a 12 to hit,
and the DM rolls a 99 on percentile dice.
The bottom 99 in the "12" column of the
20-Roll Binomial Table corresponds to 15
hits, so at least this many hit. There are six
99's in the column; six minus one is five, so
this is the number of extra attempts that
must still be determined. To deal with
these five, the DM consults the 5-Roll
Binomial Table and rolls a 92, indicating 
4 more hits for a total of 19.  Drilled 
with arrows, Francis hires himself out as a 
hatrack. 

How the system works
Where did all these numbers come
from? From calculating probabilities. A
probability is nothing more than a number
between 0 and 1, inclusive. If something
has a probability of 0, there is no chance
that it will happen, while a probability of 1
means the event is certain to happen.
Numbers? in between indicate percentage 
chances; thus, .75 means an event will 
have a 75% chance of occurring.  If 2 
events are independent (such as 2 rolls 
of a die), the chance that both will occur is 
the product of the 2 probabilities. 

Now let?s consider a concrete example:
The "10" column on the 5-Roll Binomial
Table. There is an 11 in 20 chance of rolling
a 10 or better on a 20-sided die, which
corresponds to a .55 probability. Since the
five rolls are independent, the probability
of them all being 10 or higher is found by
multiplying them together: .55 multiplied
by itself five times equals .0503, or about
5%. Subtracting 5 from 100 yields 95; this
is the number that must be rolled on percentile
dice to simulate rolling 10 or
higher five consecutive times on a 20-sided
die. This checks with the table, where 95
is the entry that corresponds to five hits
when the ?to hit? number (or saving
throw, or whatever) is 10.

Since there is a 55% chance of a hit,
there is a 45% chance of a miss. So, the
chance of missing the first hit and succeeding
on the other four is
.45 × .55× .55 × .55 × .55, which is .0412.
Because the single miss is equally likely to
happen on the 2nd, 3rd, 4th, or 5th try, 
the total chance for 4 successful hits is 
5 x .0412, or about 21%.  Subtracting this 
from 95 gives 74.  

There are 10 ways to hit 3 times 
and miss twice; this corresponds to a 34%
chance, and 74-34 = 40, which is one off 
the listed value.  The reason for the discrepancy 
is round-off error; we have been 
rounding during the calculations, while 
the tables were computed 1st and then 
rounded. 

DM: Boy these binomal tables are great!  
Now, I can run encounters with entire 
armies! 

Francis Jr: (a new 1st-level cleric): Oh, 
goody.

5-ROLL BINOMIAL TABLE

--	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
5	23	41	56	67	76	83	88	92	95	97	98	99	99	99	99	99	99	99	99
4	02	08	16	26	37	47	57	66	74	81	87	91	95	97	98	99	99	99	99
3	00	01	03	06	10	16	24	32	41	50	59	68	76	84	90	94	97	99	99
02	00	00	00	01	02	03	05	09	13	19	26	34	43	53	63	74	84	92	98
1	00	00	00	00	00	00	01	01	02	03	05	08	12	17	24	33	44	59	77

10-ROLL BINOMIAL TABLE

--	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
10	40	65	80	89	94	97	99	99	99	99	99	99	99	99	99	99	99	99	99
9	09	26	46	62	76	85	91	95	98	99	99	99	99	99	99	99	99	99	99
8	01	07	18	32	47	62	74	83	90	95	97	99	99	99	99	99	99	99	99
7	00	01	05	12	22	35	49	62	73	83	90	95	97	99	99	99	99	99	99
6	00	00	01	03	08	15	25	37	50	62	74	83	91	95	98	99	99	99	99
5	00	00	00	01	02	05	09	17	26	38	50	63	75	85	92	97	99	99	99
4	00	00	00	00	00	01	03	05	10	17	27	38	51	65	78	88	95	99	99
3	00	00	00	00	00	00	00	01	03	05	10	17	26	38	53	68	82	93	99
2	00	00	00	00	00	00	00	00	00	01	02	05	09	15	24	38	34	74	91
1	00	00	00	00	00	00	00	00	00	00	00	01	01	03	06	11	20	35	60

20-ROLL BINOMIAL TABLE

--	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
20	64	88	96	99	99	99	99	99	99	99	99	99	99	99	99	99	99	99	99
19	26	61	82	93	98	99	99	99	99	99	99	99	99	99	99	99	99	99	99
18	08	32	60	79	91	96	99	99	99	99	99	99	99	99	99	99	99	99	99
17	02	13	35	59	77	89	96	98	99	99	99	99	99	99	99	99	99	99	99
16	00	04	17	37	59	76	88	95	98	99	99	99	99	99	99	99	99	99	99
15	00	01	07	20	38	58	75	87	94	98	99	99	99	99	99	99	99	99	99
14	00	00	02	09	21	39	58	75	87	94	98	99	99	99	99	99	99	99	99
13	00	00	01	03	10	23	40	58	75	87	94	98	99	99	99	99	99	99	99
12	00	00	00	01	04	11	24	40	59	75	87	94	98	99	99	99	99	99	99
11	00	00	00	00	01	05	12	24	41	59	75	87	95	98	99	99	99	99	99
10	00	00	00	00	00	02	05	13	25	41	59	76	88	95	99	99	99	99	99
9	00	00	00	00	00	01	02	06	13	25	41	60	76	89	96	99	99	99	99
8	00	00	00	00	00	00	01	02	06	13	25	42	60	77	90	97	99	99	99
7	00	00	00	00	00	00	00	01	02	06	13	25	42	61	79	91	98	99	99
6	00	00	00	00	00	00	00	00	01	02	06	13	25	42	62	80	93	99	99
5	00	00	00	00	00	00	00	00	00	01	02	05	12	24	41	63	83	96	99
4	00	00	00	00	00	00	00	00	00	00	00	02	04	11	23	41	65	87	98
3	00	00	00	00	00	00	00	00	00	00	00	00	01	03	09	21	40	68	92
2	00	00	00	00	00	00	00	00	00	00	00	00	00	01	02	07	18	39	74
1	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	01	04	12	36