Zombie Dice Statistical Analysis: Getting the most brains without getting shotgunned

Some days it is really fun and interesting being the admin and author of a blog.  I am slightly compulsive about checking my traffic stats throughout the day.  This lets me know how many people have visited my blog, and if they got there by clicking some link somewhere else out on the Internet.  It also lets me know what search terms were used to find my blog.  While most of these are pretty predictable, every now and then I see something really silly that led to my site.   But it also gives me a keen insight into what people are interested in, and oddly enough, the search term that keeps leading to me over and over again is “how to get the most brains in zombie dice” and “strategies for zombie dice”.   I’ve reviewed Zombie Dice in the past, but I have never delved into the math involved with getting the most brains, and at your request, I’m going to dig into that today.

Your collection of dice 13 dice used in Zombie Dice.

Your collection of dice 13 dice used in Zombie Dice.

There are two major pieces of information that you should look at when playing zombie dice:  What are the conditions in the cup, and what are your odds of rolling a shotgun?  If you break each one of those down, your decision becomes much easier.  The first thing to consider is what is the likelihood of rolling a brain, shotgun, or feet on each color of die.

Odds of rolling each
symbol
on a single die

Brain Shotgun Feet
Red 16.7% 50% 33.4%
Yellow 33.4% 33.4% 33.4%
Green 50% 16.7% 33.4%

As you can see, your gut instincts for what symbols are likely to be rolled are largely justified.  It makes sense that you are so much more likely to roll a shotgun on a red die and a brain on a green die.  It’s just common sense.  However, what is more interesting is the likelihood of rolling one or more shotguns on a single roll of three dice.

Odds of rolling 1 shotgun on a single roll of three dice,
depending on the colors of the dice rolled.

R & R R & Y R & G Y & Y Y & G G & G
Red 50%
Yellow 44% 38.9% 33.3%
Green 38.9% 33.3% 27.8% 27.8% 22.2% 16.6%

Odds of rolling 2 shotguns on a single roll of three dice,
depending on the colors of the dice rolled.

R & R R & Y R & G Y & Y Y & G G & G
Red 25%
Yellow 19.5% 15.12% 11.1%
Green 15.12% 11.1% 7.7% 7.7% 4.9% 2.7%

Odds of rolling 3 shotguns on a single roll of three dice,
depending on the colors of the dice rolled.

R & R R & Y R & G Y & Y Y & G G & G
Red 12.5%
Yellow 8.6 % 5.8% 3.69%
Green 5.8% 3.69% 2.1% 2.1% 1.09% 0.45%

These statistics reveal some really interesting things.  Firstly that your odds are never any worse than 50/50.  That’s better than any casino in the country, and thus means that the odds of you rolling a shotgun is stacked in your favor, and the odds of you rolling multiples in a single roll drops drastically.   But what I find equally fascinating is when the same percentage appears in the table in two different spots.  For example, the odds of you rolling a single shotgun blast is the same regardless of if you are rolling all yellow dice, or if you are rolling a red, a yellow, and a green.  This can be easily expressed as:

R + G = 2Y.

If you apply some simple integers into those variables it becomes easy to figure your odds in much the same way that an individual counts cards.  For example lets substitute in the value 5 for the R and 1 for the G.  5 + 1 = 2Y.  That would force Y to be 3, a value nicely in the middle.   From this we can make some conclusions based upon dice combinations:

If we have three red dice, the combined value is 15, and is our worst odds possible.

If we have three green dice, the combined value is 3, and is our best odds possible.

If we have three yellow dice, the combined value is 9, and is the middle of the pack.

If we have one red (5), one yellow (3), and one green (1) the combined value is 9: the same odds as having three yellow dice.  So you know the odds of rolling a shotgun is the same on both sets of dice.

What does this mean?  It means that you can quickly access the amount of risk a roll will have, and if it is above or below the threshold of risk you would like to maintain.  For instance, if you would like to maintain that you only roll when you have a 25% chance of getting a shotgun, you would want to keep the value of your roll to four or less (aka 3 greens (3), or a yellow(2) and two greens(2).  However 2 yellows and a green would put you over your desired threshold of risk.)

A view inside the cup. There are 8 dice still inside the cup (2 red, 5 green, and 1 yellow) so the value inside the cup is 18.

But that is assuming that we all know exactly the hand we are going to draw out of the cup each time we decide to roll, and of course we don’t (although you might have an idea based upon how many feet you had rolled previously).  These odds can also be evaluated based upon a simple count of the point system (because points work much better than coming up with the exact percentage when dealing with that many variables).   The game starts out with 33 points in the cup (3 red (5 points each), 4 yellow (3 points each), and 6 green (1 point each)), and the percentage chance of drawing at least one red die is 23.07% when starting.  As you draw dice out of the cup your point value in the cup will change, by subtracting the value drawn each time. For example, if you draw a red, a yellow, and a green die on your first turn the point value of your cub will drop from 33 to 24, by subtracting 9 which was the value associated with the dice you just drew.  There are now only 10 dice left in the cup, and the percentage chance that you will draw another red die on your next draw is 20%.   Notice that by drawing a red, and making the point value of the cup lower, the likelihood of you drawing another red has decreased.

But this leaves one big gaping whole in our plans to access risk, we have to access that risk in the context of how many dice are left.  This can easily be expressed as:

( R+G+Y)/T = Ri

or
(Reds + Greens + Yellows)/Total number of Dice in cup = Risk

So what does this really mean?  It gives a weight to each die left in the cup that is between 1 and 5.  The higher this number the more risk associated with taking it.  For example, in the picture above there are 5 greens, 2 reds, and 1 yellow die in the cup.  This totals to 18 points.  Given that there are 8 dice in the cup, we can access the risk of each die to be 2.25.  This is saying that each dice is less risky than a yellow, but nearly twice the risk of a green.

18/8 = 2.25

Now I know what you are thinking, “But I’m not going to be able to figure out 2.25 in my head, on the fly”.  Well, maybe not, but you can easily determine that 8 goes into 18 twice, but not three times, so you know the risk is between 2 and 3.  Less risky than yellow, but twice as risky as a green.    In retrospect, 2.25 is actually a slightly better risk assessment than you initially start each round with.

33/13 = 2.53

When you apply this to our previous evaluation of risk on a per roll basis, that means that your first roll of the game is always going to have a risk level of 7.6 (3 times the risk assessment of a single die, or 3 x 2.53).  Slightly less then the our “middle of the pack” number of 9, but much higher then our ideal risk of 3.

So, that is a whole lot of math and logic, and possibly way more than the folks who search for “how to get the most brains” wanted to know.  However, I think it’s a very interesting look at a very simplistic game.  So now, go have fun eating brains, and good luck with not getting shot gunned.

This entry was posted by The_Null_Entry on Tuesday, October 25th, 2011 at 8:52 am and is filed under Random Things that Fall out of my head . You can follow any responses to this entry through the RSS 2.0 feed. You can skip to the end and leave a response. Pinging is currently not allowed.

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