Dice for Non-Mathematicians - 301

Besties
The the previous two posts, we've talked about how adding dice together changes the weight of different kinds of rolls.  But what if we AREN'T adding the dice together? What if people roll several dice and keep only the best roll?

Let's assume you have a game where a player rolls 1d6 and scores a point if it's a 4 or higher. What are his chances? 50%: [1,2,3] don't score and [4,5,6] do. Same as flipping a coin, right? So what would happen to his chances if you let him roll 3 times and keep the highest roll? Would his odds of scoring improve?



You probably want to say, "yes." And you'd be right. But to understand why, let's play the same game with a coin flip instead of the die. (The odds work out the same, and it'll be simpler to describe with a coin).  The possible outcomes for one flip are:
Tails (T)
Heads (H)

So if flipping "Heads" scores a point, you have a 1-in-2 chance of scoring.

Now, if you get to flip that coin twice instead, you'd expect to see an improvement in your odds.  If we lay out the possible outcomes we get:
T,T
H,T
T,T
T,H

So there is only one way to NOT score (T,T) and 3 ways to score. That means you have a 3-in-4 chance to score (i.e. 75%). With three coins, the only way NOT to score is (T,T,T)... But there are 7 ways to score. So your chances of scoring are 7-in-8 (87.5%).

Lock 
So, let's take it a step farther... Let's say you score for EVERY head you flip.  So, if tails are 0 points, and heads are one point, your possible flips are:
T,T,T = 0 pts
T,T,H = 1 pt
T,H,T = 1 pt
H,T,T = 1 pt
T,H,H = 2 pts
H,T,H = 2 pts
H,H,T = 2 pts
H,H,H = 3 pts

Without using any fancy math, we can work out that there is a 1-in-8 chance to score 3 points.  And it is just as likely to score 0.


Try Try Again
What if you can flip your 3 coins keep the heads, and then re-flip the tails?  Well, that layers on an extra level of complexity, because each possible outcome from the first round branches into more possible outcomes.

But it may not be as complex as it seems.

So, if someone flipped all heads in the first round, then they wouldn't reflip anything. So, let's keep moving.
The only possible outcome is: H,H,H (all 3 are locked; nothing is flipped the second time): 3 points.

If someone flipped 2 Heads in the first round, they would only be reflipping one coin... which means the possible outcomes are.
H,H,H (2 locked, reflip is Heads): 3 points
H,H,T (2 locked, reflip is Tails): 2 points

First round one-pointers gets a little messier.  H,T,T (where we lock-in the H) can be reflipped as:
H,T,T: 1 points
H,T,H: 2 points
H,H,T: 2 points
H,H,H: 3 points

Half the time you re-flip a first-round one-pointer you will wind up with 2 points.  A quarter of the time you will end with 1 point.  A quarter of the time you will end with 3 points.

The table for first round no-pointers would look the exact same as the first round of flips (since you're just reflipping everything.

Which means, if we stack up every variation of flipping, locking Heads and reflipping once we get:

First round 3 pointers:
3 points

First round 2 pointers:
2 points
3 points

First round 1 pointers
1 points
2 points
2 points
3 points

First round 0 pointers:
0 points
1 points
1 points
1 points
2 points
2 points
2 points
3 points

Total number of outcomes: 17
Chances of 3 points: 4-in-17
Chances of 2 points: 6-in-17
Chances of 1 point: 6-in-17
Chances of 0 points: 1-in-17

Bottom Line:
If you've come this far, then you have gone well beyond the basics of dice mechanics.  We could pick apart a 1,000 other examples and case studies, but hopefully you're feeling empowered to work out your own specifics for yourself.

At the end of the day, (as we've said all along) you simply need to think carefully about every possible road to get to the outcome you're interested, and see if it jives with your vision for the game. Diagramming will tell you the odds, but your vision for the game (and the information you glean from playtests) will tell you if those are the right odds.