654

Strategy

If you are like me, you’ve been up at night ever since Austin thinking, “What does the long term distribution of 6-5-4 rolls looks like?” or “How likely am I to qualify at all?” I’m happy to share we’ll no longer be losing sleep over these existential questions.

To get an idea of what game results are most likely, I wrote some code to play 5 million consecutive games of 6-5-4. This is a common strategy in statistics called Monte Carlo simulation, which is useful when there are too many complex situations to list.

The game’s flow is mostly straightforward with saving die, but I choose to proceed with the strategy/choice of stopping whenever I get my first qualified hand, no matter the turn. This means that my simulated games will stop on roll 1 or 2 even if I have the lowest possible score of 2 (roll: 6-5-4-1-1). My rationale is that it’s better to make your opponent do the harder task of qualifying in fewer rolls, even if the end score wasn’t that impressive.

Code

Under the hood it’s a series of if() statements (also know as if-thens) where I record if we roll numbers of interest:

# on game number ii

# first check for a 6
  if(any(rll == 6)){
    
    qual6[ii] <- TRUE # record successful 6
    
    # then check for a 5 if we have a 6
    if(any(rll == 5)){
      
      qual5[ii] <- TRUE # record successful 5
      
      # then check for a 4 if we have a 5
      if(any(rll == 4)){
        
        qual4[ii] <- TRUE # record successful 4
        
      }
    } 
  } 

Basically all of these things have to be true, in that order for us to get a qualified score. If we don’t get a valid, qualified score from the initial roll, then repeat the rolling process at most 2 more times while saving important die (e.g. 6s, 5s, and 4s) along the way. I record a few pieces of information about each game. Here is a look at the first 5 games I simulated. For simplicity I chose to label it “Turn 4 | Score 0” when I failed to qualify during my 3-roll turn.

Qual6	Qual5	Qual4	Score	Qual_Turn	Roll1	Roll2	Roll3	Result
TRUE	TRUE	TRUE	6	1	65442	NA	NA	Turn 1 | Score 6
TRUE	TRUE	TRUE	5	2	64322	65432	NA	Turn 2 | Score 5
TRUE	TRUE	TRUE	6	2	54211	65541	NA	Turn 2 | Score 6
TRUE	FALSE	FALSE	0	4	55522	66321	66442	Turn 4 | Score 0
TRUE	TRUE	TRUE	3	3	44322	55521	65421	Turn 3 | Score 3

This gives us 34 possible results. The absolute best hand is Turn 1 | Score 12, which comes from a single roll of 6-5-4-6-6. The worst qualifying hand is Turn 3 | Score 2, which happens with 6-5-4-1-1 on your third roll. The worst hand is a non-qualifying hand which I call Turn 4 | Score 0.

Results

After playing 5,000,000 games, long term trends start to become clear. Here is a breakdown of the relative likelihood of each of those 34 game results.

There is a 46.01% chance of busting, which leaves a 53.99% chance of rolling a qualifying score in your 3-roll turn. Each individual result is unlikely to happen by itself, but since you technically have 33 different ways to score, it ends up being a pretty fair game to score at.

We can also look at the probability of qualifying on each roll (or eventually busting on “turn 4”):

You have a 15.82% chance of scoring on your first roll, and a 35.64% chance of qualifying on your first two rolls.