a question of dice mechanics

[raga]

Hey all, I've got a question here about a game I'm trying to develop and I'm curious if some of the more math talented could chime in about the following dice mechanics. That being said, here goes

option 1) players roll 1D6 per trait involved in the test, each D6 that is 4+ is considered a sucsess, the others are failures. Does this stay at 50% success/failure rate no matter how many dice are used?

option 2) players roll 1D6 per trait involved in the test, depending on the level of the trait a success is one of the following 3+/4+/5+ . what are the percentage chances of success if you add more dice than 1, ie % chance of success for 6D6 with success at 3+?

I would love to hear input from anyone, and if this doesn't make sense just ask me to explain please.

peace,
mike

[JimmyB]

Since rolling two dice does not actually affect either one, no matter how many you roll so long as you consider the dice seperately the percentage chances will be the same, you're just more likely overall to get success/failure.

For 4+ the chance will always be 50%
5+ it'll be 33%
3+ it'll be 66%

If you roll multiple dice then it will make a difference to the overall odds though.

2 dice will be the percentage squared, i.e.
4+ will be 50%*50%* of both coming up successes, or 25% overall
There'll be the same chance of both coming up failures, and a 50% chance of one success out of the two.

[Thierry Michel]

players roll 1D6 per trait involved in the test, depending on the level of the trait a success is one of the following 3+/4+/5+ . what are the percentage chances of success if you add more dice than 1, ie % chance of success for 6D6 with success at 3+?

OK, so you set a task number (TN) between 1 and 6. Your probability of failure on one die is then (TN-1)/6.  Now, the probability of failing all of N dice is ((TN-1)/6)^N. Thus the probability of getting at least one success is 1 - ((TN-1)/6)^N.

In your example, 1 - ((3-1)/6)^6 ~99.9 %

The probs of getting 1 or more successes are only slightly more complex (and a bit tedious to compute - use a spreadsheet).

-----------------------
# success   Probs
0................. 0.1%
1 .................1.6%
2.................8.2%
3.................21.9%
4.................32.9%
5.................26.3%
6.................8.8%
---------------------

Hope this helps.

[Mike Holmes]

Here are some threads that deal with probability. Note that some people have posted bad math (myself included, potentially), so make sure that you've got the correct formula before employing it.

http://www.indie-rpgs.com/viewtopic.php?t=875
http://www.indie-rpgs.com/viewtopic.php?t=3095
http://www.indie-rpgs.com/viewtopic.php?t=6203

Mike

[xechnao]

2 dice will be the percentage squared, i.e.
4+ will be 50%*50%* of both coming up successes, or 25% overall
There'll be the same chance of both coming up failures, and a 50% chance of one success out of the two.

So can we say you have a 50%+25%=75% chance of at least one success out of the two?

[Walt Freitag]

Mike (raga), your questions aren't very clear to me. In option 1, are you asking about the chance of rolling success on at least one die, rolling success on more than half the dice, rolling success on half the dice or more, or not rolling any failures?

However, in an attempt to be as helpful as possible, here are tablulations of probabilities for different outcomes of d6 rolls using various criteria for success per die and various numbers of dice. The probabilities are rounded; "0.00% chance" indicates a chance of less than 1/200th of 1%, or less than 1 in 20,000, but not actually zero.

Code:

SUCCESS ON ROLL OF 6 ONLY (OR 1 ONLY)

1 die

1 success,    0 failures:    16.67%
0 successes,  1 failure:     83.33%

2 dice

2 successes,  0 failures:     2.78%
1 success,    1 failure:     27.78%
0 successes,  2 failures:    69.44%

3 dice

3 successes,  0 failures:     0.46%
2 successes,  1 failure:      6.94%
1 success,    2 failures:    34.72%
0 successes,  3 failures:    57.87%

4 dice

4 successes,  0 failures:     0.08%
3 successes,  1 failure:      1.54%
2 successes,  2 failures:    11.57%
1 success,    3 failures:    38.58%
0 successes,  4 failures:    48.23%

5 dice

5 successes,  0 failures:     0.01%
4 successes,  1 failure:      0.32%
3 successes,  2 failures:     3.22%
2 successes,  3 failures:    16.08%
1 success,    4 failures:    40.19%
0 successes,  5 failures:    40.19%

6 dice

6 successes,  0 failures:     0.00%
5 successes,  1 failure:      0.06%
4 successes,  2 failures:     0.80%
3 successes,  3 failures:     5.36%
2 successes,  4 failures:    20.09%
1 success,    5 failures:    40.19%
0 successes,  6 failures:    33.49%

7 dice

7 successes,  0 failures:     0.00%
6 successes,  1 failure:      0.01%
5 successes,  2 failures:     0.19%
4 successes,  3 failures:     1.56%
3 successes,  4 failures:     7.81%
2 successes,  5 failures:    23.44%
1 success,    6 failures:    39.07%
0 successes,  7 failures:    27.91%

8 dice

8 successes,  0 failures:     0.00%
7 successes,  1 failure:      0.00%
6 successes,  2 failures:     0.04%
5 successes,  3 failures:     0.42%
4 successes,  4 failures:     2.60%
3 successes,  5 failures:    10.42%
2 successes,  6 failures:    26.05%
1 success,    7 failures:    37.21%
0 successes,  8 failures:    23.26%

9 dice

9 successes,  0 failures:     0.00%
8 successes,  1 failure:      0.00%
7 successes,  2 failures:     0.01%
6 successes,  3 failures:     0.10%
5 successes,  4 failures:     0.78%
4 successes,  5 failures:     3.91%
3 successes,  6 failures:    13.02%
2 successes,  7 failures:    27.91%
1 success,    8 failures:    34.89%
0 successes,  9 failures:    19.38%

10 dice

10 successes, 0 failures:     0.00%
9 successes,  1 failure:      0.00%
8 successes,  2 failures:     0.00%
7 successes,  3 failures:     0.02%
6 successes,  4 failures:     0.22%
5 successes,  5 failures:     1.30%
4 successes,  6 failures:     5.43%
3 successes,  7 failures:    15.50%
2 successes,  8 failures:    29.07%
1 success,    9 failures:    32.30%
0 successes,  10 failures:   16.15%

11 dice

11 successes, 0 failures:     0.00%
10 successes, 1 failure:      0.00%
9 successes,  2 failures:     0.00%
8 successes,  3 failures:     0.01%
7 successes,  4 failures:     0.06%
6 successes,  5 failures:     0.40%
5 successes,  6 failures:     1.99%
4 successes,  7 failures:     7.11%
3 successes,  8 failures:    17.77%
2 successes,  9 failures:    29.61%
1 success,    10 failures:   29.61%
0 successes,  11 failures:   13.46%

12 dice

12 successes, 0 failures:     0.00%
11 successes, 1 failure:      0.00%
10 successes, 2 failures:     0.00%
9 successes,  3 failures:     0.00%
8 successes,  4 failures:     0.01%
7 successes,  5 failures:     0.11%
6 successes,  6 failures:     0.66%
5 successes,  7 failures:     2.84%
4 successes,  8 failures:     8.88%
3 successes,  9 failures:    19.74%
2 successes,  10 failures:   29.61%
1 success,    11 failures:   26.92%
0 successes,  12 failures:   11.22%


SUCCESS ON ROLL OF 5-6 (OR 1-2)

1 die

1 success,    0 failures:    33.33%
0 successes,  1 failure:     66.67%

2 dice

2 successes,  0 failures:    11.11%
1 success,    1 failure:     44.44%
0 successes,  2 failures:    44.44%

3 dice

3 successes,  0 failures:     3.70%
2 successes,  1 failure:     22.22%
1 success,    2 failures:    44.44%
0 successes,  3 failures:    29.63%

4 dice

4 successes,  0 failures:     1.23%
3 successes,  1 failure:      9.88%
2 successes,  2 failures:    29.63%
1 success,    3 failures:    39.51%
0 successes,  4 failures:    19.75%

5 dice

5 successes,  0 failures:     0.41%
4 successes,  1 failure:      4.12%
3 successes,  2 failures:    16.46%
2 successes,  3 failures:    32.92%
1 success,    4 failures:    32.92%
0 successes,  5 failures:    13.17%

6 dice

6 successes,  0 failures:     0.14%
5 successes,  1 failure:      1.65%
4 successes,  2 failures:     8.23%
3 successes,  3 failures:    21.95%
2 successes,  4 failures:    32.92%
1 success,    5 failures:    26.34%
0 successes,  6 failures:     8.78%

7 dice

7 successes,  0 failures:     0.05%
6 successes,  1 failure:      0.64%
5 successes,  2 failures:     3.84%
4 successes,  3 failures:    12.80%
3 successes,  4 failures:    25.61%
2 successes,  5 failures:    30.73%
1 success,    6 failures:    20.48%
0 successes,  7 failures:     5.85%

8 dice

8 successes,  0 failures:     0.02%
7 successes,  1 failure:      0.24%
6 successes,  2 failures:     1.71%
5 successes,  3 failures:     6.83%
4 successes,  4 failures:    17.07%
3 successes,  5 failures:    27.31%
2 successes,  6 failures:    27.31%
1 success,    7 failures:    15.61%
0 successes,  8 failures:     3.90%

9 dice

9 successes,  0 failures:     0.01%
8 successes,  1 failure:      0.09%
7 successes,  2 failures:     0.73%
6 successes,  3 failures:     3.41%
5 successes,  4 failures:    10.24%
4 successes,  5 failures:    20.48%
3 successes,  6 failures:    27.31%
2 successes,  7 failures:    23.41%
1 success,    8 failures:    11.71%
0 successes,  9 failures:     2.60%

10 dice

10 successes, 0 failures:     0.00%
9 successes,  1 failure:      0.03%
8 successes,  2 failures:     0.30%
7 successes,  3 failures:     1.63%
6 successes,  4 failures:     5.69%
5 successes,  5 failures:    13.66%
4 successes,  6 failures:    22.76%
3 successes,  7 failures:    26.01%
2 successes,  8 failures:    19.51%
1 success,    9 failures:     8.67%
0 successes,  10 failures:    1.73%

11 dice

11 successes, 0 failures:     0.00%
10 successes, 1 failure:      0.01%
9 successes,  2 failures:     0.12%
8 successes,  3 failures:     0.75%
7 successes,  4 failures:     2.98%
6 successes,  5 failures:     8.35%
5 successes,  6 failures:    16.69%
4 successes,  7 failures:    23.84%
3 successes,  8 failures:    23.84%
2 successes,  9 failures:    15.90%
1 success,    10 failures:    6.36%
0 successes,  11 failures:    1.16%

12 dice

12 successes, 0 failures:     0.00%
11 successes, 1 failure:      0.00%
10 successes, 2 failures:     0.05%
9 successes,  3 failures:     0.33%
8 successes,  4 failures:     1.49%
7 successes,  5 failures:     4.77%
6 successes,  6 failures:    11.13%
5 successes,  7 failures:    19.08%
4 successes,  8 failures:    23.84%
3 successes,  9 failures:    21.20%
2 successes,  10 failures:   12.72%
1 success,    11 failures:    4.62%
0 successes,  12 failures:    0.77%


SUCCESS ON ROLL OF 4-6 (OR 1-3)

1 die

1 success,    0 failures:    50.00%
0 successes,  1 failure:     50.00%

2 dice

2 successes,  0 failures:    25.00%
1 success,    1 failure:     50.00%
0 successes,  2 failures:    25.00%

3 dice

3 successes,  0 failures:    12.50%
2 successes,  1 failure:     37.50%
1 success,    2 failures:    37.50%
0 successes,  3 failures:    12.50%

4 dice

4 successes,  0 failures:     6.25%
3 successes,  1 failure:     25.00%
2 successes,  2 failures:    37.50%
1 success,    3 failures:    25.00%
0 successes,  4 failures:     6.25%

5 dice

5 successes,  0 failures:     3.12%
4 successes,  1 failure:     15.62%
3 successes,  2 failures:    31.25%
2 successes,  3 failures:    31.25%
1 success,    4 failures:    15.62%
0 successes,  5 failures:     3.12%

6 dice

6 successes,  0 failures:     1.56%
5 successes,  1 failure:      9.38%
4 successes,  2 failures:    23.44%
3 successes,  3 failures:    31.25%
2 successes,  4 failures:    23.44%
1 success,    5 failures:     9.38%
0 successes,  6 failures:     1.56%

7 dice

7 successes,  0 failures:     0.78%
6 successes,  1 failure:      5.47%
5 successes,  2 failures:    16.41%
4 successes,  3 failures:    27.34%
3 successes,  4 failures:    27.34%
2 successes,  5 failures:    16.41%
1 success,    6 failures:     5.47%
0 successes,  7 failures:     0.78%

8 dice

8 successes,  0 failures:     0.39%
7 successes,  1 failure:      3.12%
6 successes,  2 failures:    10.94%
5 successes,  3 failures:    21.88%
4 successes,  4 failures:    27.34%
3 successes,  5 failures:    21.88%
2 successes,  6 failures:    10.94%
1 success,    7 failures:     3.12%
0 successes,  8 failures:     0.39%

9 dice

9 successes,  0 failures:     0.20%
8 successes,  1 failure:      1.76%
7 successes,  2 failures:     7.03%
6 successes,  3 failures:    16.41%
5 successes,  4 failures:    24.61%
4 successes,  5 failures:    24.61%
3 successes,  6 failures:    16.41%
2 successes,  7 failures:     7.03%
1 success,    8 failures:     1.76%
0 successes,  9 failures:     0.20%

10 dice

10 successes, 0 failures:     0.10%
9 successes,  1 failure:      0.98%
8 successes,  2 failures:     4.39%
7 successes,  3 failures:    11.72%
6 successes,  4 failures:    20.51%
5 successes,  5 failures:    24.61%
4 successes,  6 failures:    20.51%
3 successes,  7 failures:    11.72%
2 successes,  8 failures:     4.39%
1 success,    9 failures:     0.98%
0 successes,  10 failures:    0.10%

11 dice

11 successes, 0 failures:     0.05%
10 successes, 1 failure:      0.54%
9 successes,  2 failures:     2.69%
8 successes,  3 failures:     8.06%
7 successes,  4 failures:    16.11%
6 successes,  5 failures:    22.56%
5 successes,  6 failures:    22.56%
4 successes,  7 failures:    16.11%
3 successes,  8 failures:     8.06%
2 successes,  9 failures:     2.69%
1 success,    10 failures:    0.54%
0 successes,  11 failures:    0.05%

12 dice

12 successes, 0 failures:     0.02%
11 successes, 1 failure:      0.29%
10 successes, 2 failures:     1.61%
9 successes,  3 failures:     5.37%
8 successes,  4 failures:    12.08%
7 successes,  5 failures:    19.34%
6 successes,  6 failures:    22.56%
5 successes,  7 failures:    19.34%
4 successes,  8 failures:    12.08%
3 successes,  9 failures:     5.37%
2 successes,  10 failures:    1.61%
1 success,    11 failures:    0.29%
0 successes,  12 failures:    0.02%


SUCCESS ON ROLL OF 3-6 (OR 1-4)

1 die

1 success,    0 failures:    66.67%
0 successes,  1 failure:     33.33%

2 dice

2 successes,  0 failures:    44.44%
1 success,    1 failure:     44.44%
0 successes,  2 failures:    11.11%

3 dice

3 successes,  0 failures:    29.63%
2 successes,  1 failure:     44.44%
1 success,    2 failures:    22.22%
0 successes,  3 failures:     3.70%

4 dice

4 successes,  0 failures:    19.75%
3 successes,  1 failure:     39.51%
2 successes,  2 failures:    29.63%
1 success,    3 failures:     9.88%
0 successes,  4 failures:     1.23%

5 dice

5 successes,  0 failures:    13.17%
4 successes,  1 failure:     32.92%
3 successes,  2 failures:    32.92%
2 successes,  3 failures:    16.46%
1 success,    4 failures:     4.12%
0 successes,  5 failures:     0.41%

6 dice

6 successes,  0 failures:     8.78%
5 successes,  1 failure:     26.34%
4 successes,  2 failures:    32.92%
3 successes,  3 failures:    21.95%
2 successes,  4 failures:     8.23%
1 success,    5 failures:     1.65%
0 successes,  6 failures:     0.14%

7 dice

7 successes,  0 failures:     5.85%
6 successes,  1 failure:     20.48%
5 successes,  2 failures:    30.73%
4 successes,  3 failures:    25.61%
3 successes,  4 failures:    12.80%
2 successes,  5 failures:     3.84%
1 success,    6 failures:     0.64%
0 successes,  7 failures:     0.05%

8 dice

8 successes,  0 failures:     3.90%
7 successes,  1 failure:     15.61%
6 successes,  2 failures:    27.31%
5 successes,  3 failures:    27.31%
4 successes,  4 failures:    17.07%
3 successes,  5 failures:     6.83%
2 successes,  6 failures:     1.71%
1 success,    7 failures:     0.24%
0 successes,  8 failures:     0.02%

9 dice

9 successes,  0 failures:     2.60%
8 successes,  1 failure:     11.71%
7 successes,  2 failures:    23.41%
6 successes,  3 failures:    27.31%
5 successes,  4 failures:    20.48%
4 successes,  5 failures:    10.24%
3 successes,  6 failures:     3.41%
2 successes,  7 failures:     0.73%
1 success,    8 failures:     0.09%
0 successes,  9 failures:     0.01%

10 dice

10 successes, 0 failures:     1.73%
9 successes,  1 failure:      8.67%
8 successes,  2 failures:    19.51%
7 successes,  3 failures:    26.01%
6 successes,  4 failures:    22.76%
5 successes,  5 failures:    13.66%
4 successes,  6 failures:     5.69%
3 successes,  7 failures:     1.63%
2 successes,  8 failures:     0.30%
1 success,    9 failures:     0.03%
0 successes,  10 failures:    0.00%

11 dice

11 successes, 0 failures:     1.16%
10 successes, 1 failure:      6.36%
9 successes,  2 failures:    15.90%
8 successes,  3 failures:    23.84%
7 successes,  4 failures:    23.84%
6 successes,  5 failures:    16.69%
5 successes,  6 failures:     8.35%
4 successes,  7 failures:     2.98%
3 successes,  8 failures:     0.75%
2 successes,  9 failures:     0.12%
1 success,    10 failures:    0.01%
0 successes,  11 failures:    0.00%

12 dice

12 successes, 0 failures:     0.77%
11 successes, 1 failure:      4.62%
10 successes, 2 failures:    12.72%
9 successes,  3 failures:    21.20%
8 successes,  4 failures:    23.84%
7 successes,  5 failures:    19.08%
6 successes,  6 failures:    11.13%
5 successes,  7 failures:     4.77%
4 successes,  8 failures:     1.49%
3 successes,  9 failures:     0.33%
2 successes,  10 failures:    0.05%
1 success,    11 failures:    0.00%
0 successes,  12 failures:    0.00%


SUCCESS ON ROLL OF 2-6 (OR 1-5)

1 die

1 success,    0 failures:    83.33%
0 successes,  1 failure:     16.67%

2 dice

2 successes,  0 failures:    69.44%
1 success,    1 failure:     27.78%
0 successes,  2 failures:     2.78%

3 dice

3 successes,  0 failures:    57.87%
2 successes,  1 failure:     34.72%
1 success,    2 failures:     6.94%
0 successes,  3 failures:     0.46%

4 dice

4 successes,  0 failures:    48.23%
3 successes,  1 failure:     38.58%
2 successes,  2 failures:    11.57%
1 success,    3 failures:     1.54%
0 successes,  4 failures:     0.08%

5 dice

5 successes,  0 failures:    40.19%
4 successes,  1 failure:     40.19%
3 successes,  2 failures:    16.08%
2 successes,  3 failures:     3.22%
1 success,    4 failures:     0.32%
0 successes,  5 failures:     0.01%

6 dice

6 successes,  0 failures:    33.49%
5 successes,  1 failure:     40.19%
4 successes,  2 failures:    20.09%
3 successes,  3 failures:     5.36%
2 successes,  4 failures:     0.80%
1 success,    5 failures:     0.06%
0 successes,  6 failures:     0.00%

7 dice

7 successes,  0 failures:    27.91%
6 successes,  1 failure:     39.07%
5 successes,  2 failures:    23.44%
4 successes,  3 failures:     7.81%
3 successes,  4 failures:     1.56%
2 successes,  5 failures:     0.19%
1 success,    6 failures:     0.01%
0 successes,  7 failures:     0.00%

8 dice

8 successes,  0 failures:    23.26%
7 successes,  1 failure:     37.21%
6 successes,  2 failures:    26.05%
5 successes,  3 failures:    10.42%
4 successes,  4 failures:     2.60%
3 successes,  5 failures:     0.42%
2 successes,  6 failures:     0.04%
1 success,    7 failures:     0.00%
0 successes,  8 failures:     0.00%

9 dice

9 successes,  0 failures:    19.38%
8 successes,  1 failure:     34.89%
7 successes,  2 failures:    27.91%
6 successes,  3 failures:    13.02%
5 successes,  4 failures:     3.91%
4 successes,  5 failures:     0.78%
3 successes,  6 failures:     0.10%
2 successes,  7 failures:     0.01%
1 success,    8 failures:     0.00%
0 successes,  9 failures:     0.00%

10 dice

10 successes, 0 failures:    16.15%
9 successes,  1 failure:     32.30%
8 successes,  2 failures:    29.07%
7 successes,  3 failures:    15.50%
6 successes,  4 failures:     5.43%
5 successes,  5 failures:     1.30%
4 successes,  6 failures:     0.22%
3 successes,  7 failures:     0.02%
2 successes,  8 failures:     0.00%
1 success,    9 failures:     0.00%
0 successes,  10 failures:    0.00%

11 dice

11 successes, 0 failures:    13.46%
10 successes, 1 failure:     29.61%
9 successes,  2 failures:    29.61%
8 successes,  3 failures:    17.77%
7 successes,  4 failures:     7.11%
6 successes,  5 failures:     1.99%
5 successes,  6 failures:     0.40%
4 successes,  7 failures:     0.06%
3 successes,  8 failures:     0.01%
2 successes,  9 failures:     0.00%
1 success,    10 failures:    0.00%
0 successes,  11 failures:    0.00%

12 dice

12 successes, 0 failures:    11.22%
11 successes, 1 failure:     26.92%
10 successes, 2 failures:    29.61%
9 successes,  3 failures:    19.74%
8 successes,  4 failures:     8.88%
7 successes,  5 failures:     2.84%
6 successes,  6 failures:     0.66%
5 successes,  7 failures:     0.11%
4 successes,  8 failures:     0.01%
3 successes,  9 failures:     0.00%
2 successes,  10 failures:    0.00%
1 success,    11 failures:    0.00%
0 successes,  12 failures:    0.00%

- Walt

[raga]

Wow! thanks for the quick replies people. I realise my question probably could have been more specific, sorry.

Walt... my man... thanks for the huge printout, in the end I think that was actually the kind of data I wanted to see.

Mike. thanks for the links. I need to figure out how to do this on my own.

Thierry. thanks for the calcs. its all helping me to understand probability better.

Many thanks to all. this will help me pin down the mechanics I want to use. When I have more I'll post it.

peace,
mike