So I see it fairly simple.

On the first roll, I have 100% chance of changing one focus to a crit.

If I reroll all 4, there is 75% chance that each dice comes up as a hit, crit or focus and there is a relatively small chance of getting blanks on all 4 of the dice - BUT - it is there.

I would take the sure thing, change one to a crit, and reroll the other 3.

It's situational. If you NEED that one crit as a game winds down (and you know you can push it through), sure take it. But otherwise, rerolling that eyeball with the others yields (on average) slightly higher results. I worked out the odds previously with RAC's ability.

It's like baseball. Under normal circumstances with zero outs and a man on first, sacrifice bunting him to second is a poor idea as it lowers the total run expectancy. However, if you need just ONE run to tie or take the lead, "manufacturing a run" is more of a "sure thing".

To be specific, in the original poster's scenario, there's a 57.8% of rolling an additional eyeball on 3 dice. Not overwhelming, but more often than not, Ezra will be staring at a second eyeball he can't convert.

[EDIT: that should read 57.8% of rolling at least an additional eyeball on 3 dice]

The reasoning behind option 2 is completely ignoring the concept of dependent outcomes. In this example, four foci have already been rolled. This does not magically impact the results of the next four dice.

Do *you* have a PhD in physics from Cambridge?

Wow. This card again. I respect the loyalty you feel in jumping to XBear's defense, but his credentials are totally irrelevant here. I am aware that the original poster is both highly educated and intelligent, but in this matter, he has thoroughly lost sight of fundamentals.

Tell you what, take this to a STATISTICS professor and enjoy his/her response.

option 2, reroll 3 dice: dice rolls are independent from one another, rolling 8 dice all together is the same as rolling 8 dice one at a time, or 4 dice twice. the probability of getting 5 or more focuses when rolling 8 red dice is about 3%. Since we already know that 4 results are all focuses, the probability of having at least another focus in the next 4 results is about 3%.

That is not how this works. You're looking at two different events:

1. Getting 5+ focus results when rolling 8 dice

2. Getting 5+ focus results when rolling 8 dice, given I have already rolled 4 dice and gotten 4 focus results.

If you have already rolled 4 focus results, and you are wondering whether rolling another 4 dice will get you at least 1 focus, you're basically looking at the event "Getting at least 1 Focus on 4 dice", which is a pretty good chance.

THISSSSSS

The reasoning behind option 2 is completely ignoring the concept of dependent outcomes. In this example, four foci have already been rolled. This does not magically impact the results of the next four dice.

what dependent outcomes? the dice do not know you've stopped after 4 results to consider. all 8 results are independent rolls. the 4 dice you roll do not depend on the 4 dice before. the first 4 dice results do not impact the results of the next four dice. if it were so, you could not apply the binomial distribution to all 8, for example, as the binomial distribution applies to independent outcomes.

You're making my argument for me in everything save your conclusion. I highlighted the key point in your post above. I need some clarification on your perspective. Let's take 8 dice. I manually flip 4 of them to eyeballs. I roll the next 4. Are you claiming that I'll only roll at least one eyeball ~ 3% of the time? Or is this magically exempt because I didn't actually roll the first 4?

you can roll one die at a time or all 4 together, it doesn't matter. so imagine you roll one at a time. imagine you're rolling the 4th dice. somebody jumps in and says, oh you have a 25% chance of rolling a focus there. but it's your 4th dice and you rolled 3 focuses. the chance of rolling one focus on one die is 25%, the chance of rolling 4 focuses in 4 dice is 0.4%. so who's right? in a way, both statements are right. there is no contradiction in the sense that you're measuring the probabilities of different events. one is the chance of one result with one die, the other is the chance of 4 results with 4 dice. you're not choosing the chance you prefer, you're just saying, OK, of course I know that the chance of getting a focus with a single dice, even the 4th dice in this example, is 25%. However, I also know that the chance of this different event, of having 4 consecutive focuses, is only 0.4%, so I'm not holding my breath to get a 4th focus. it can and does sometimes happen, but not very often.

You're temporally unbound, here. For those of us that experience linear time, it works like this:

If you have four dice in hand, ready to roll, your chance of getting 4 eyeballs is low (0.25^4 ~= 0.4%). However, if you roll out three eyeballs, then pick up a fourth die, given that you already have three, your chance for a fourth is 25%. Yeah, it's unlikely that you ended up here, but it's also pretty unlikely that you got those three eyeballs beforehand. Going from 3 to 4 eyes given you have the three already is just the bog-standard 25%.

Ignore probabilities. Dice are sentient and like to screw you over (except mine, so please be kind to me at the tournament tomorrow please dice?), so if you re-roll all four you won't get an eyeball

I know this isn't true, but it FEELS true, and that's something isn't it.

Following your example and talking to my dice more might just work...

Storing my dice with hit results and evade results face up has helped.

To put it another way: If I flip a coin 50 times and get heads every time; what are the odds the next flip will be heads? 50/50.

This is actually how probabilities work mathematically. Personally I think the real mechanics are scarier than if magic really did give you +5% bonus for rolling incorrectly, because in reality, there is nothing you can do to make the dice roll better.

It is total.

Random.

Luck. Luck? Luck.