Now, it's been a bit since I took probability in College, but I believe the Math in the first post isn't correct.

The best way to determine the odds of a success/fail roll is using the Binomial Distribution Formula. If you look up the formula on Wikipedia, it seems pretty complicated but really isn't. The first part is read as N choose K, and the second part is pretty self explanatory. N is how many trials we will be performing, and K is how many successes we need.

Thus, since we have 1 roll plus two rerolls, and we only need 1 success, we would say 3 choose 1. 3 choose 1 doesn't seem like math, but it is. You can plug 3 choose 1 exactly into google and it will spit out your answer. Since the probability of a double hit occurring is 1/8, or .125, we would say: 3 choose 1 * (.125)^1 * (1-.875)^2. That would equal 28.71% chance of having a double hit. Of course this assumes that you are rerolling everything until that sweet double.

Now you can use that new probability to plug in the same exact formula for each dice you need. For instance, in your example you go with 4 Red Dice. Thus, for exactly 1 double hit: 4 choose 1 * (.2871)^1 * (1-.2871)^3 or 41.61%. You can do this for as many successes that you want. Example for two successes: 4 choose 2 * (.2871)^2 * (1-.2871)^2 = 25.13%.

This formula, along with expected value can be used to solve almost any damage related question for upgrades and dice in this game.

Actually I've been using the binomial distribution formula the whole way through, for what it's worth.

The problem with your math is that rerolls are conditional on the status of the earlier dice rolls. We'll want to keep some but reroll others. We can't roll one dice 3 times and just take the best result.

In this case, rerolls are indistinguishable from re-rolling a totally new dice. It doesn't matter what roll you actually get the double hit because all that matters is the end result.

You're right though, I did make a mistake in my math. I forgot to add in the probabilities of rolling 2 or 3 double hits since we will be rolling the dice up to 3 times. Even know we will stop rolling the die as soon as we get a double, we still need to include those possible future dice rolls to get the correct answer.

As for Upgrades, expected damage is definitely the way to do it. I think it's very interesting how Ordnance experts, a mere 4 point upgrade, can give the MC30 a 1.25 increase in damage. That is a very good value I think.

Thank you for this. Combining Vader, Needa, TRC and Overload Pulse made a huge difference in my regular game against my regular opponent. (we're a gaming group that totals 2 lol) Both his torpedo shrimp and C90 were oneshotted hehe

Can I ask why it always results in an increase?

A blank black dice has a 75% chance of coming up with increased damage.

A black dice with 1 hit on it, has the exact same chance of coming up no damage, as it does increasing damage, both 25%.

So I fully understand that rerolling a blank black dice has a 3 in 4 chance of rolling damage, (50% 1 damage, 25% 2 damage), and rerolls are great for that, but every time I see any type of discussion on rerolling black dice that have a damage result, everyone always says your chance increases, yet your chance does not increase, 50% chance remaining the same, 25% of going up to 2 damage, 25% of going down to zero damage.

I do not understand, and I am not trying to start an argument, I'd love to have it explained is all.

This is why Screed is better when only rolling 2-4 black dice, than Vader, Vader comes into his own once you get passed that guaranteed chance of not losing damage, or increasing damage Screed gives.

Can I ask why it always results in an increase?

A blank black dice has a 75% chance of coming up with increased damage.

A black dice with 1 hit on it, has the exact same chance of coming up no damage, as it does increasing damage, both 25%.

So I fully understand that rerolling a blank black dice has a 3 in 4 chance of rolling damage, (50% 1 damage, 25% 2 damage), and rerolls are great for that, but every time I see any type of discussion on rerolling black dice that have a damage result, everyone always says your chance increases, yet your chance does not increase, 50% chance remaining the same, 25% of going up to 2 damage, 25% of going down to zero damage.

I do not understand, and I am not trying to start an argument, I'd love to have it explained is all.

This is why Screed is better when only rolling 2-4 black dice, than Vader, Vader comes into his own once you get passed that guaranteed chance of not losing damage, or increasing damage Screed gives.

I don't think anybody claimed that rerolling your single hits on black dice improves damage. It increases your chances of getting a HIT+CRIT while leaving damage the same. This can in fact improve your average damage by proccing effects like Assault Concussion Missiles or Assault Proton Torpedoes, however, but it does not change the average damage provided by the dice themselves.

Here's your average black dice roll:

50% odds of 1 damage = +0.5 average damage

25% odds of 2 damage (hit+crit) = +0.5 average damage

25% odds of a blank - +0 average damage.

Thus you get an average of 1 for a regular non-rerollable roll.

Once you factor in rerolling blanks, you take that 25% nothing and turn it into the same results again, so you effectively add:

25% chance of blank, thus reroll * 50% odds of new roll is 1 damage = +0.125 average damage

25% chance of blank, thus reroll * 25% odds of new roll is 2 damage = +0.125 average damage

25% chance of blank, thus reroll * 25% odds of new roll is blank = +0 average damage

So rerolling only the blanks adds 0.25 average damage per dice to the earlier result of 1 average damage = 1.25 average damage. You have a 31.25% chance per dice of it being a crit (25% chance + (25%*25%)).

You can reroll the blanks AND hits to get similar results, in which case we'd only "keep" the hits+crits from the original roll (0.5 average damage)

75% chance of rerolling * 50% odds new roll is 1 damage = +0.375 average damage

75% chance of rerolling * 25% odds new roll is 2 damage = +0.375 average damage

75% chance of rerolling * 25% odds new roll is blank = +0 average damage

So rerolling the blanks AND the single hits adds 0.75 average damage per dice to the earlier result of 0.5 average damage = 1.25 average damage. You have a 43.75% chance per dice of it being a crit (25% chance + (75%*25%)).

The only way that rerolling the single hits produces more total damage on the dice themselves is if you're stacking two rerolls together. In that case, it simply comes down to you have a second chance to reroll blanks to a better result so it's best to use the crit-fishing reroll style (rerolling all non-hit+crit results) for your first reroll and then you can decide for your second reroll how aggressive you want to be with it. I can do the math on that one too but it takes longer so I'd rather not provided it made sense to everyone.

I absolutely love math posts and this dimension of the game. Thanks for posting this up. I'm not super big on the Imperials at the moment, but I love the challenge of figuring out how to make ships that are slightly out of the meta or less frequently used actually work.

Keep it up!

Yeah I'm beginning to think I should just get some kind of "Snipafist blathers about probability at people/takes orders" thread going and just link to it in my signature.

This is the kind of stuff I would have LOVED to have read up on when I started wargaming but before I understood probability better or found a binomial distribution calculator.

In somewhat-related news, I ran Vader with two ISD-Is tonight and those rerolls with ISD-Is are INSANE at close range.

I'm not going to claim it's a tier-one combination but WOW. Combine with Intel Officer and season to taste.

Can I ask why it always results in an increase?

A blank black dice has a 75% chance of coming up with increased damage.

A black dice with 1 hit on it, has the exact same chance of coming up no damage, as it does increasing damage, both 25%.

So I fully understand that rerolling a blank black dice has a 3 in 4 chance of rolling damage, (50% 1 damage, 25% 2 damage), and rerolls are great for that, but every time I see any type of discussion on rerolling black dice that have a damage result, everyone always says your chance increases, yet your chance does not increase, 50% chance remaining the same, 25% of going up to 2 damage, 25% of going down to zero damage.

I do not understand, and I am not trying to start an argument, I'd love to have it explained is all.

This is why Screed is better when only rolling 2-4 black dice, than Vader, Vader comes into his own once you get passed that guaranteed chance of not losing damage, or increasing damage Screed gives.

I don't think anybody claimed that rerolling your single hits on black dice improves damage. It increases your chances of getting a HIT+CRIT while leaving damage the same. This can in fact improve your average damage by proccing effects like Assault Concussion Missiles or Assault Proton Torpedoes, however, but it does not change the average damage provided by the dice themselves.

Here's your average black dice roll:

50% odds of 1 damage = +0.5 average damage

25% odds of 2 damage (hit+crit) = +0.5 average damage

25% odds of a blank - +0 average damage.

Thus you get an average of 1 for a regular non-rerollable roll.

Once you factor in rerolling blanks, you take that 25% nothing and turn it into the same results again, so you effectively add:

25% chance of blank, thus reroll * 50% odds of new roll is 1 damage = +0.125 average damage

25% chance of blank, thus reroll * 25% odds of new roll is 2 damage = +0.125 average damage

25% chance of blank, thus reroll * 25% odds of new roll is blank = +0 average damage

So rerolling only the blanks adds 0.25 average damage per dice to the earlier result of 1 average damage = 1.25 average damage. You have a 31.25% chance per dice of it being a crit (25% chance + (25%*25%)).

You can reroll the blanks AND hits to get similar results, in which case we'd only "keep" the hits+crits from the original roll (0.5 average damage)

75% chance of rerolling * 50% odds new roll is 1 damage = +0.375 average damage

75% chance of rerolling * 25% odds new roll is 2 damage = +0.375 average damage

75% chance of rerolling * 25% odds new roll is blank = +0 average damage

So rerolling the blanks AND the single hits adds 0.75 average damage per dice to the earlier result of 0.5 average damage = 1.25 average damage. You have a 43.75% chance per dice of it being a crit (25% chance + (75%*25%)).

The only way that rerolling the single hits produces more total damage on the dice themselves is if you're stacking two rerolls together. In that case, it simply comes down to you have a second chance to reroll blanks to a better result so it's best to use the crit-fishing reroll style (rerolling all non-hit+crit results) for your first reroll and then you can decide for your second reroll how aggressive you want to be with it. I can do the math on that one too but it takes longer so I'd rather not provided it made sense to everyone.

Thank you.

"In somewhat-related news, I ran Vader with two ISD-Is tonight and those rerolls with ISD-Is are INSANE at close range."

Please help me in the Devastator thread, I'm taking grenades over there putting up a defense for Vader...

Btw, did you combine with OE??? Would love to see list... cheers

"In somewhat-related news, I ran Vader with two ISD-Is tonight and those rerolls with ISD-Is are INSANE at close range."

Please help me in the Devastator thread, I'm taking grenades over there putting up a defense for Vader...

Btw, did you combine with OE??? Would love to see list... cheers

I've avoided the Devastator thread because my personal opinion is Devastator is overpriced. FFG doesn't seem to price "increase chances of dying in return for 1 or 2 more dice" abilities all that well. My current opinion on Vader is that viewing him as a damage buff is only half-right. He helps you get closer to the exact dice outcome you want for the situation. He really shines on ISDs and Raiders thus far (because you can reroll the blue dice looking for a single accuracy in addition to the black dice that can be rerolled by Ordnance Experts for cheaper), although I've had some success with him on VSDs that had Slaved Turrets + Intel Officers. In general he likes ships that get one big attack because you don't generally want to be spending more than one defense token a turn (...usually) and so he also pairs well with Intel Officers, which also like ships with fewer but more powerful attacks. Anyways, here's what I ran for funsies...

Faction: Galactic Empire

Points: 398/400

Commander: Darth Vader

Assault Objective: Advanced Gunnery

Defense Objective: Contested Outpost

Navigation Objective: Dangerous Territory

[ flagship ] Imperial I-Class Star Destroyer (110 points)

- Darth Vader ( 36 points)

- Relentless ( 3 points)

- Intel Officer ( 7 points)

Imperial I-Class Star Destroyer (110 points)

- Avenger ( 5 points)

- Intel Officer ( 7 points)

Raider-I Class Corvette (44 points)

Raider-I Class Corvette (44 points)

4 TIE Fighter Squadrons ( 32 points)

"In somewhat-related news, I ran Vader with two ISD-Is tonight and those rerolls with ISD-Is are INSANE at close range."

Please help me in the Devastator thread, I'm taking grenades over there putting up a defense for Vader...

But Vader combines so much better with Avenger and its motly crew!

Boo, Vader.

As Snip already pointed out Vader helps you kill but he also helps you die. His point cost is only 1/3 of why he is too expensive. He is expensive in opportunity cost (what those extra precious points could have bought you), he is expensive in FP cost and he is expensive to use. If you are using Motti, he is free to use, Ozzel is virtually free to use (simply multiplies the effect of a command you would have thrown anyway), Tarkin is free to use, Screed costs you a die but if you are using him for his ability his cost is likely resulting in a net-gain. Vader subtracts some of your ability to defend yourself effectively and therefore has a decision tree that effectively cuts down his use-cases. It doesn't matter how his re-roll maths out if using said re-roll will likely leave you **** to the wind against a focused attack.

I will take leading shots, OE or other dice modifying secondary effects (Tarkin CF tokens, Screed dice mod, and any combination of the preceeding)

Like Devastator he is too expensive, both in face cost and real cost.

I don't dispute that Vader is an expensive commander, but I've been using him pretty consistently this month (we did a more laid-back league at my FLGS and I was tired of mainlining Screed most of the time as I'd been playing him for nearly two months straight to get my Adepticon fleet ready) and finding he's really not that bad. He's primarily underwhelming with Gladiators, I've found (as the Ordnance Experts are an easy choice because you've got black dice everywhere and it's not worth flipping a defense token you don't get multiples of to reroll the 2 red dice). He does fine with Raiders, ISDs, and VSDs. He particularly likes being paired with Intel Officers.

This is more about probability and average improvements than "Boo, Vader," though.

Btw. Anyone:

TRC Needa and General Tagg to refresh your evade on T3?

Btw. Anyone:

TRC Needa and General Tagg to refresh your evade on T3?

I mentioned that exact combination in the newest flotillas/wave 3 thread and I expect it to be somewhat popular, especially early on once we're figuring out Tagge.