A question of probability [Archive] - Wizards Community

A question of probability [Archive] - Wizards Community

Post/Author/DateTime Post
Prophet of Chaos
12-19-06, 12:09 PM What are the odds of a character being born with 6 18s (or of a PC just rolling those stats for a char)? You know...18 Str, Dex, Con, Int, Wis & Cha? I seem to remember someone mentioning this, years ago, but I can't find a thread like that anywhere around here. So uh...is there a mathematician in the house?
Rahlious
12-19-06, 12:19 PM It's very low. I'll post the calculation details in a moment.
Ocalus
12-19-06, 12:21 PM What are the odds of a character being born with 6 18s (or of a PC just rolling those stats for a char)? You know...18 Str, Dex, Con, Int, Wis & Cha? I seem to remember someone mentioning this, years ago, but I can't find a thread like that anywhere around here. So uh...is there a mathematician in the house?

Using the 3d6 method the odds are 1:101,559,956,668,416 (or 1 in 101 trillion for those who don't want to count the digits).

The odds of getting an 18 using 3d6 is 1 in 216, so the odds of getting six 18's using that method is 216*216*216*216*216*216.

As a basis for comparison, the odds of winning a 6/49 lottery (1 in 10 billion) are literally more than ten-thousand times better.
StarFyre
12-19-06, 12:26 PM via 4d6 method: I'm getting an answer of 4.033X10^-11

I did it as: 1/6*1/6*1/6*6/6 (chance of getting 3 6s and on the 4th dice getting any #).

then I multiplied by 4, since with 4 dice, there are 4 ways this could happen.

Then multiplied exponent 6 since it wouldbe an "AND" condition from probability for each result getting some combination to yield 18.

Sanjay
Rahlious
12-19-06, 12:26 PM When you roll 4d6 drop the lowest, there are 6^4 possible rolls if you think of the dice as distinct.

6^4 = 1296

Only 24 of those rolls give a stat of 18. So when you roll your first stat, the chance of getting an 18 is:

24 / 1296 = 0.01852 = 1.852 %

The chance of doing it six times in a row is:
0.01852^6 = 4.033*10^(-9) %
Witch
12-19-06, 12:30 PM then I multiplied by 4, since with 4 dice, there are 4 ways this could happen.
5 ways. You could get 4 6's.
StarFyre
12-19-06, 01:10 PM There are 4 ways.

Dice 1, 2, 3, 4

if 1, 2, 3 all get 6..then dice 4 can be any #
or
2, 3, 4 could be 6, and 1 could be any #
the same with 2, or 3

basically I did:

[4(1/6)(1/6)(1/6)(6/6)]^6

Sanjay
Solaris
12-19-06, 01:46 PM No, no, no. Ignore everyone there except Ocalus.

The probability of rolling an 18 with the best 3 of 4d6 is 21/1296. The probability of rolling 6 of them is therefore (21/1296)^6. That's all there is to it. Ocalus gave the correct answer for 3d6 using the same method.
Solaris
12-19-06, 03:58 PM The number of ways to roll three 6s and one less than 6 is
4!/(3!1!) (1/6)^3 (5/6)^1 = 4 * 1/6^3 * 5/6 = 20/1296. To this we add
the number of ways of rolling four 6s, which is (1/6)^4 = 1/1296.
20/1296 + 1/1296 = 21/1296.
Witch
12-19-06, 04:01 PM Oh, yeah. Solaris is correct.
Ocalus
12-19-06, 04:25 PM No, no, no. Ignore everyone there except Ocalus.

The probability of rolling an 18 with the best 3 of 4d6 is 21/1296. The probability of rolling 6 of them is therefore (21/1296)^6. That's all there is to it. Ocalus gave the correct answer for 3d6 using the same method.

Which I think works out to 1:55,171,016,309 (1 in 55 billion) odds, though someone more adept at math than I could work it out more accurately I think.
Solaris
12-19-06, 04:32 PM Which I think works out to 1:55,171,016,309 (1 in 55 billion) odds, though someone more adept at math than I could work it out more accurately I think.

For the best 3 of 4d6 the odds of rolling all 18s are 1 in 6499837226778624/117649, or about 1 in 55,247,704,841. Yeah, 1 in 55 billion is good enough :)
StoneStokes
12-19-06, 06:13 PM There are 21 ways to roll 18 using 4d6-drop-lowest.

The fast way to compute this is as follows:
Either all four come up 6 (one way), or exactly three come up 6 (how many ways can this happen?
There are four choices for the low die, and five choices the number it reads. Thus, 20 ways.
For a total of 21.

For those not yet convinced, I will list them. If I have too many, you should be able to point out where I have reported the same possibility twice. If I have too few, you should be able to tell me a possibility I have forgotten:

1. 6666
2. 6665
3. 6664
4. 6663
5. 6662
6. 6661
7. 6656
8. 6646
9. 6636
10. 6626
11. 6616
12. 6566
13. 6466
14. 6366
15. 6266
16. 6166
17. 5666
18. 4666
19. 3666
20. 2666
21. 1666

I have computed (analytically, and exactly) the number of ways each score may be rolled out of 1296, using 4d6-drop-lowest. There is a separate thread entitled "Ability Score Probabilities."

Cheers,
StoneStokes
Prophet of Chaos
12-19-06, 06:44 PM Wow. 1 in 55 billion. I really didn't think the odds would be that high. Nevertheless gentlemen, I appreciate the prompt response. Thank you!
Ashok_Lal
12-19-06, 08:46 PM the odds increase dramaticly if you cheat:)
Ocalus
12-20-06, 02:57 PM Wow. 1 in 55 billion. I really didn't think the odds would be that high. Nevertheless gentlemen, I appreciate the prompt response. Thank you!

What is even more amazing is not that the odds are 1:55 billion but rather the staggeringly large number of people who have claim to have done it fairly and without cheating. ;)
High Octane
12-20-06, 03:25 PM Its 1 in about 54 billion.

Edit: Rule of thumb, if they said they rolled 6 18s, assume they are lying. Sure its possible one day somebody will get them when no one is watching, but you are knocking down around 54 billion others that were cheaters for that one.

Those are odds I can live with.

Which is why Im not exalted.
Solaris
12-20-06, 03:45 PM Its 1 in about 54 billion.A bit over 55 billion, as posted above.
High Octane
12-20-06, 04:45 PM A bit over 55 billion, as posted above.

Even galileo was a touch off.

Post/Author/DateTime	Post
Prophet of Chaos 12-19-06, 12:09 PM	What are the odds of a character being born with 6 18s (or of a PC just rolling those stats for a char)? You know...18 Str, Dex, Con, Int, Wis & Cha? I seem to remember someone mentioning this, years ago, but I can't find a thread like that anywhere around here. So uh...is there a mathematician in the house?
Rahlious 12-19-06, 12:19 PM	It's very low. I'll post the calculation details in a moment.
Ocalus 12-19-06, 12:21 PM	What are the odds of a character being born with 6 18s (or of a PC just rolling those stats for a char)? You know...18 Str, Dex, Con, Int, Wis & Cha? I seem to remember someone mentioning this, years ago, but I can't find a thread like that anywhere around here. So uh...is there a mathematician in the house? Using the 3d6 method the odds are 1:101,559,956,668,416 (or 1 in 101 trillion for those who don't want to count the digits). The odds of getting an 18 using 3d6 is 1 in 216, so the odds of getting six 18's using that method is 216216216216216*216. As a basis for comparison, the odds of winning a 6/49 lottery (1 in 10 billion) are literally more than ten-thousand times better.
StarFyre 12-19-06, 12:26 PM	via 4d6 method: I'm getting an answer of 4.033X10^-11 I did it as: 1/61/61/6*6/6 (chance of getting 3 6s and on the 4th dice getting any #). then I multiplied by 4, since with 4 dice, there are 4 ways this could happen. Then multiplied exponent 6 since it wouldbe an "AND" condition from probability for each result getting some combination to yield 18. Sanjay
Rahlious 12-19-06, 12:26 PM	When you roll 4d6 drop the lowest, there are 6^4 possible rolls if you think of the dice as distinct. 6^4 = 1296 Only 24 of those rolls give a stat of 18. So when you roll your first stat, the chance of getting an 18 is: 24 / 1296 = 0.01852 = 1.852 % The chance of doing it six times in a row is: 0.01852^6 = 4.033*10^(-9) %
Witch 12-19-06, 12:30 PM	then I multiplied by 4, since with 4 dice, there are 4 ways this could happen. 5 ways. You could get 4 6's.
StarFyre 12-19-06, 01:10 PM	There are 4 ways. Dice 1, 2, 3, 4 if 1, 2, 3 all get 6..then dice 4 can be any # or 2, 3, 4 could be 6, and 1 could be any # the same with 2, or 3 basically I did: [4(1/6)(1/6)(1/6)(6/6)]^6 Sanjay
Solaris 12-19-06, 01:46 PM	No, no, no. Ignore everyone there except Ocalus. The probability of rolling an 18 with the best 3 of 4d6 is 21/1296. The probability of rolling 6 of them is therefore (21/1296)^6. That's all there is to it. Ocalus gave the correct answer for 3d6 using the same method.
Solaris 12-19-06, 03:58 PM	The number of ways to roll three 6s and one less than 6 is 4!/(3!1!) (1/6)^3 (5/6)^1 = 4 * 1/6^3 * 5/6 = 20/1296. To this we add the number of ways of rolling four 6s, which is (1/6)^4 = 1/1296. 20/1296 + 1/1296 = 21/1296.
Witch 12-19-06, 04:01 PM	Oh, yeah. Solaris is correct.
Ocalus 12-19-06, 04:25 PM	No, no, no. Ignore everyone there except Ocalus. The probability of rolling an 18 with the best 3 of 4d6 is 21/1296. The probability of rolling 6 of them is therefore (21/1296)^6. That's all there is to it. Ocalus gave the correct answer for 3d6 using the same method. Which I think works out to 1:55,171,016,309 (1 in 55 billion) odds, though someone more adept at math than I could work it out more accurately I think.
Solaris 12-19-06, 04:32 PM	Which I think works out to 1:55,171,016,309 (1 in 55 billion) odds, though someone more adept at math than I could work it out more accurately I think. For the best 3 of 4d6 the odds of rolling all 18s are 1 in 6499837226778624/117649, or about 1 in 55,247,704,841. Yeah, 1 in 55 billion is good enough :)
StoneStokes 12-19-06, 06:13 PM	There are 21 ways to roll 18 using 4d6-drop-lowest. The fast way to compute this is as follows: Either all four come up 6 (one way), or exactly three come up 6 (how many ways can this happen? There are four choices for the low die, and five choices the number it reads. Thus, 20 ways. For a total of 21. For those not yet convinced, I will list them. If I have too many, you should be able to point out where I have reported the same possibility twice. If I have too few, you should be able to tell me a possibility I have forgotten: 1. 6666 2. 6665 3. 6664 4. 6663 5. 6662 6. 6661 7. 6656 8. 6646 9. 6636 10. 6626 11. 6616 12. 6566 13. 6466 14. 6366 15. 6266 16. 6166 17. 5666 18. 4666 19. 3666 20. 2666 21. 1666 I have computed (analytically, and exactly) the number of ways each score may be rolled out of 1296, using 4d6-drop-lowest. There is a separate thread entitled "Ability Score Probabilities." Cheers, StoneStokes
Prophet of Chaos 12-19-06, 06:44 PM	Wow. 1 in 55 billion. I really didn't think the odds would be that high. Nevertheless gentlemen, I appreciate the prompt response. Thank you!
Ashok_Lal 12-19-06, 08:46 PM	the odds increase dramaticly if you cheat:)
Ocalus 12-20-06, 02:57 PM	Wow. 1 in 55 billion. I really didn't think the odds would be that high. Nevertheless gentlemen, I appreciate the prompt response. Thank you! What is even more amazing is not that the odds are 1:55 billion but rather the staggeringly large number of people who have claim to have done it fairly and without cheating. ;)
High Octane 12-20-06, 03:25 PM	Its 1 in about 54 billion. Edit: Rule of thumb, if they said they rolled 6 18s, assume they are lying. Sure its possible one day somebody will get them when no one is watching, but you are knocking down around 54 billion others that were cheaters for that one. Those are odds I can live with. Which is why Im not exalted.
Solaris 12-20-06, 03:45 PM	Its 1 in about 54 billion.A bit over 55 billion, as posted above.
High Octane 12-20-06, 04:45 PM	A bit over 55 billion, as posted above. Even galileo was a touch off.