So, with one yellow proficiency die, your chances of rolling a Triumph are 1 in 12. What are the odds of getting at least one Triumph if you're rolling two yellows? And is there a set probability increase with each additional yellow die you roll? (I'm not talking about success vs. failure here; just the raw probability that one of those yellows will come up a Triumph.)

Bonus points if you can show me the mathematical formula for determining the added chances of getting a Triumph with each additional yellow die. I know this is something I learned once upon a time...

You might enjoy http://anydice.com/

Chance for one triumph on two dice = chance of rolling it on dice one + chance of rolling it on dice two - chance of rolling it on both

= 1/12 + 1/12 - 1/144 = 23/144

Going with the reverse event is the easiest way: The chance to roll at least one triumph is equal to 100% minus the chance to not roll a single one. Every yellow die has a chance of 11/12 to not roll a triumph, so we get this:
p(triumph)=1-(11/12)^n
where n is the amount of yellow dice you roll, giving us:
- about 16% for 2
- about 23% for 3
- about 29,4% for 4
and so on.

P(at least 1 triumph)=1 - P(no triumph)^n

Where P is the probability of an event and n is the number of die.

So if you have three die it is

1 - (11/12)^3=397/1728

43 minutes ago, Darzil said:

Chance for one triumph on two dice = chance of rolling it on dice one + chance of rolling it on dice two - chance of rolling it on both

= 1/12 + 1/12 - 1/144 = 23/144

The OP's question was "at least one Triumph", not precisely one Triumph. Thus the correct answer to his question is 25/144. A hair over 1 in 6.

The chance to pop at least one Triumph increases by 1/12 per each yellow die added to the pool. So with 3 yellows you have a slightly better than 1 in 4 chance, and with 4 yellows you have slightly better than a 1 in 3 chance.

Excellent. Shado's answer is what I was looking for, but I love to see how all this is calculated. Looks like with the maximum of five yellows in a pool, you'd have a 5/12 chance to pop at least one Triumph, meaning about 42%.

I suppose with a maxed out five ranks in skill, a 6 in a characteristic, and a Destiny Point spend, you could get to a 50% probability to get at least one Triumph. That'd be where it peaked, unless you had cybernetics to go to a 7 and used something like True Aim. Thanks, all!

It's slightly better than 5/12, because rare double and very rare triple results also qualify as "at least one". But you can round down to 5/12 for convenience. If I worked out the math it might be 43 or 44%. With 5 ranks, Dedication to 6 green, cybernetics to 7 green, DP and True Aim you could have 7 yellows. You can't get any more greens except through upgrades, and to get an 8th yellow you'd need 2 upgrades (one to get the 8th green and another for the 8th yellow). I don't know how that could be achieved.

1 hour ago, ShadoWarrior said:

It's slightly better than 5/12, because rare double and very rare triple results also qualify as "at least one". But you can round down to 5/12 for convenience. If I worked out the math it might be 43 or 44%. With 5 ranks, Dedication to 6 green, cybernetics to 7 green, DP and True Aim you could have 7 yellows. You can't get any more greens except through upgrades, and to get an 8th yellow you'd need 2 upgrades (one to get the 8th green and another for the 8th yellow). I don't know how that could be achieved.

You've found it already: True Aim. Get the Sharpshooter and Mercenary Soldier specs for a total of 4 ranks in True Aim, and boom: 3 additional upgrades on top of what you've listed. Slightly silly and a bit overkill, but still.

Gives you about a 50% chance for a triumph, by the way.

Here's your True Aim, cybernetic gun monkey at work:

: 8eP 4 successes, 6 advantage, 1 Triumph

Don't forget, if you're force sensitive you can use the sense power and upgrade two more times.

1 hour ago, Genuine said:

Don't forget, if you're force sensitive you can use the sense power and upgrade two more times.

Nine yellows on a roll. I have a new goal.

1 hour ago, SavageBob said:

Nine yellows on a roll. I have a new goal.

2 Upgrades on already upgraded dice would only add 1 new die. 1 upgrade adds a green, the next turns it yellow, etc...

Here are the % of AT LEAST 1 triumph when rolling upgraded dice (Yellow)

1Y = 8.2%

2Y = 15.98%

3Y = 22.82%

4Y = 29.35%

5Y = 35.2%

% chance of at least 2 triumphs

1Y = 0% (obviously ;))

2Y = 0.72%

3Y = 1.93%

4Y = 3.76%

5Y = 6.12%

% of at least 3 triumphs

1Y/2Y = 0%

3Y = 0.056%

4Y = 0.19%

5Y = 0.51%

10 hours ago, ShadoWarrior said:

The OP's question was "at least one Triumph", not precisely one Triumph. Thus the correct answer to his question is 25/144. A hair over 1 in 6.

The chance to pop at least one Triumph increases by 1/12 per each yellow die added to the pool. So with 3 yellows you have a slightly better than 1 in 4 chance, and with 4 yellows you have slightly better than a 1 in 3 chance.

Epic Ted's answer was better, and more accurate, but mine was right too. This isn't. Because the chance of rolling a triumph on each dice overlaps, the chance that at least one is a triumph is slightly smaller than the chance one is a triumph plus the chance the other one is.

You can see this in the extreme case of rolling 12 yellow dice, the chance would not be slightly over 12/12, it'd be less than 100%.

In Arlo's, for example, you can see that chance of rolling at least one triumph on two yellows is 23/144, not 25/144.

8 hours ago, Kallabecca said:

2 Upgrades on already upgraded dice would only add 1 new die. 1 upgrade adds a green, the next turns it yellow, etc...

It would be 6 upgrades: four ranks of True Aim, plus two more upgrades from Sense. So if you have a 7 in the stat to begin with, you get to 8 yellows with all that. If you flipped a DP, too, I think you'd add a green. My mistake.

30 minutes ago, SavageBob said:

It would be 6 upgrades: four ranks of True Aim, plus two more upgrades from Sense. So if you have a 7 in the stat to begin with, you get to 8 yellows with all that. If you flipped a DP, too, I think you'd add a green. My mistake.

Huh, and it'd only take 390xp after start if you focused. You'd be crazy narrow - only good at ranged x, but still.

1. Start with any species with Ag 3 and 100 start xp
2. Career Soldier, Spec Sharpshooter - start with 2 ranks of ranged light. Spend 90 to boost agility to 5, leave 10 for talents. (Note: Starting as Soldier/Sharpshooter allows ranged light and heavy to start with two ranks, if you want to boost gunnery, start with Ace/Gunner instead).
3. 60 xp to raise ranged light to 5
4. Sharpshooter: 30(-10) points in talents.
5. Purchase Gunner for 30 points, 75 in talents
6. Purchase Merc Soldier for 40 points, 125 in talents.
7. Purchase Force Exile, 50 point
8. Sense power, 50 points
   

Is there any way to tack on an Intense Focus maneuver to get that ninth yellow?

1 hour ago, SavageBob said:

Is there any way to tack on an Intense Focus maneuver to get that ninth yellow?

Almost the cheapest way on that list would be to spend 95 xp on exile talents and grab it - only 25xp of which aren't on the line to dedication and force rank. However, it's not a ranked talent so all it will do is add a green to the 8 yellows. You'd also need to use two maneuvers every time you wanted to use it and True Aim at the same time.

You could pick it up from Ascetic with only 90xp, including the cost of a new tree, but I'm not sure that a fifth tree is worth a 5xp savings.