More probability help?

[Jeff Wilder]

Let's say that I'm in a roll-off, using d20s, against a +16 advantage. (I lose ties.) What is my chance to win, with each roll, and how do I calculate it?

I know that I can only win if I roll an 18 or higher. If I roll an 18, my opponent must roll a 1. If I roll a 19, my opponent must roll a 1 or a 2. And if I roll a 20, my opponent must roll a 1, a 2, or a 3.

The chance of 18 v. 1 is 1/400. The chance of 19 v. 1 or 2 is 1/200. The chance of 20 v. 1, 2, or 3 is 1/133.

To get the total chance, do I just add these? (If so, the chance is 1.5 percent.)

Is there a formula for figuring this out without the brute force?

Thanks for any help.

 

[Wulf Ratbane]

Is there a formula for figuring this out without the brute force?

By far the greatest difficulty in probability questions is setting up the problem correctly. And sometimes, brute force is the best way to make sure you set it up correctly.

In this case, fortunately, there aren't that many cases in the brute force selection.

There are 400 total combinations of two rolls. So just count up your winning combinations:

18 vs. 1
19 vs. 1 or 2
20 vs. 1, 2, or 3

That's 6/400.

I stand ready to be corrected....

 

[ExploderWizard]

By far the greatest difficulty in probability questions is setting up the problem correctly. And sometimes, brute force is the best way to make sure you set it up correctly.

In this case, fortunately, there aren't that many cases in the brute force selection.

There are 400 total combinations of two rolls. So just count up your winning combinations:

18 vs. 1
19 vs. 1 or 2
20 vs. 1, 2, or 3

That's 6/400.

I stand ready to be corrected....

6/400 sounds like 1.5% to me. No offense but I wouldn't bet on you.

 

[Nifft]

To get the total chance, do I just add these? (If so, the chance is 1.5 percent.)

Yep, that's exactly right.

Cheers, -- N

 

[Storminator]

As Wulf shows in his post, there are 400 cases and the number of successful cases forms a triangle:

18 vs. 1
19 vs. 1, 2
20 vs. 1, 2, 3

So for any given bonus, we need to find out how many rows in the triangle, then find out how many entries that is.

The formula for total number of entries in the triangle is wins = n*(n+1)/2 where n is the number of rows.

So we need to find the number of rows:

n = 20 - bonus - 1

Where in this case, bonus is 16 --> n = 20-16-1 = 3 --> 3*(3+1)/2 = 6

Success is then 6/400.

PS

 

[Guy Srinivasan]

AnyDice Calculator

That calculator is great.

 

[Jeff Wilder]

As Wulf shows in his post, there are 400 cases and the number of successful cases forms a triangle:

18 vs. 1
19 vs. 1, 2
20 vs. 1, 2, 3

So for any given bonus, we need to find out how many rows in the triangle, then find out how many entries that is.

The formula for total number of entries in the triangle is wins = n*(n+1)/2 where n is the number of rows.

So we need to find the number of rows:

n = 20 - bonus - 1

Where in this case, bonus is 16 --> n = 20-16-1 = 3 --> 3*(3+1)/2 = 6

Success is then 6/400.

Sweet. Thanks, all. (And especially Storminator.)

 

[Jeff Wilder]

6/400 sounds like 1.5% to me. No offense but I wouldn't bet on you.

If you can get 100-to-1, you should.

 

[weem]

AnyDice Calculator

That calculator is great.

Nice! Thanks for the link, I was looking for one not long ago.