Today, Jack Heckel (author of The Dark Lord) is joining us for the Harper Voyager Science Fair (#HVsciencefair), to discuss Probability Science. Enjoy!
Never Tell Me the Odds
by Jack Heckel
“Something is coming. Something hungry for blood.
A shadow grows on the wall behind you, swallowing you in darkness. It is almost here.
An army of troglodytes charges into the chamber!
Wait a minute. Did you hear that? That… that sound?
That didn’t come from the troglodytes. No, that… that came from something else.
-Stranger Things, Season 1, Episode 1
Thus, begins one of the most intriguing television shows of last year, Netflix’s original series: Stranger Things. A lot has been written about the show, celebrating its unique story line, the loving attention used in constructing the 1980’s suburban neighborhood it’s set in, and the amazing performances of the actors (both children and adults). But, I don’t want to talk about any of that. (Although, I would love to expound on how I believe Eleven has the potential to be this generation’s Ripley.) No, today I want to celebrate how the show uses probability as a subtle antagonist for the main characters, and the resolution of one roll of one die as an emotional unifier for the entire series. To do that, though we need to take a deep dive into some really geeky subject matter.
When we first meet the main characters of Stranger Things, they are playing Dungeons & Dragons. Mike is running the other boys through an adventure. They are near the end, and they have journeyed deep into whatever hideous dungeon Mike’s twisted imagination has dreamed up. The boys know that somewhere lurks a creature so powerfully evil it might well destroy them all: the Demogorgon. At the critical moment when the heroes come face to face with this fearsome adversary they argue about whether Will’s character should cast a fireball or a spell of protection. The protection spell is the safe play, but the fireball could defeat the monster—if it hits.
It is a moment of high drama and pure geek pathos. If you have played a roleplaying game, you probably have a memory of a similar experience, but if you have never played a roleplaying game you may not appreciate why Mike and the gang were so anxious about the outcome, or why so much depended on one roll of a die.
In some ways, Will’s roll at that moment is no different than a sporting event coming down to a single play – will the player hit the three point shot? Can the receiver catch the Hail Mary pass? A physical event, such as an athlete trying to make a play, and rolling a die may seem to be an odd comparison, but dice are used in roleplaying games to simulate the chance of an event occurring. For every action there’s a chance of failure or success, and in roleplaying games the dice capture that risk using probabilities, sometimes with ridiculous specificity. As an example, the Dungeon Master’s Guide for 1st Edition Advanced Dungeons &Dragons included a General Dungeon Dressing table which based on a die roll, randomly added anything from dried blood to mold (common) to a slimy coating on the floor, ceiling or wall. I hate to think what uncommon mold might have been.
So success on the roll would lead to victory, but there are consequences for failure. Will’s roll was going to determine for all the players whether their characters who they had played with for ten hours that day would live or die. It’s why they are so freaked out looking for the dice, and it’s why Will is so somber when he later confesses that he rolled a seven. That roll meant Demogorgon killed their characters and the campaign was over.
But, beyond the drama, what I love about the scene in Stranger Things is that it captures one of the oddest characteristics of the people (kids or adults) that play games like Dungeons & Dragons where dice feature so heavily in the outcome: they all have an instinctive understanding of probability. When the other boys are shouting at him to cast a fireball Will immediately says, “I would have to roll a 13 or higher!” It all comes down to the dice, and understanding your chance, and whether it’s “too risky” as Dustin argues while trying to convince Will to cast protection instead.
So let’s talk about probabilities. Most people think of a die as a six-sided cube, usually with pips or numbers designating each side with a number from one to six. We’ll start there. If you have an event, let’s say, deciding whether a character should go left or right, and there’s nothing particularly compelling about either direction, you start with a 50% percent chance of either direction. It’s basically the same as a coin flip. So, you assign the numbers 1, 2, and 3 to left and the numbers 4, 5, and 6 to right and roll the die. If it comes up a 2, the character goes left. Easy.
How about something more difficult? A character is engaged in a swordfight with another character. They are both evenly matched. All things being equal, we might say that the first character, let’s call him for the sake of this example… the Spaniard, needs to roll a 4, 5, or 6 to strike his opponent. His opponent, who, just for clarity, we will call something like… the Man in Black, would need a similar roll for his own attack. That would be easy with our six-sided die, but what if things aren’t so easy? What if, to make things more difficult for himself, the Spaniard is fighting with his off-hand? That could make things much harder, so instead of a 4, he needs to roll a 5, although fighting with the off-hand seems very difficult, and they are moving over rough ground, so it should probably be a 6. Sounds good, but ah, the Man in Black is using Bonetti’s Defense, so wait…we can’t make things more difficult than needing to roll a 6 with a single die, can we?
Actually, we can, but we need to roll our die more than once. We make the result more difficult by requiring us to roll a 6 on our first roll, and then we have to roll the die again and get a 3, 4, 5, or 6. What are our chances? The chance of rolling any particular result on a six-sided die is 16.67 percent, and to roll any four of six numbers gives a 66.6 percent chance, so 16.67 percent times 66.6 percent gives us a little over an 11 percent chance of success. If we start calculating for having been fatigued from climbing up the Cliffs of Madness, and the fencing styles of Capa Fero (misspelled as Capo Fero by our Spaniard) and Thibault plus Agrippa, and decide there’s a difference in fencing on the stairs versus the flat ground, our noble six-sided die doesn’t really seem to be up to the challenge. With only six possible results and these big 16.67 percent changes between them, we don’t have room for all our variables.
Fortunately, there are bunches of different dice we could use besides our six-sided cube. There’s the four-sided die with triangle faces and a 25% chance of landing on each side (which has the added benefit of serving as a great stand-in for a miniature pyramid and also as a home security countermeasure if you scatter them about your bedroom floor at night), an eight-sided with a 12.5% chance, a ten-sided with 10% chances of a side coming up (that looks so much like the eight sided die that you end up turning them around and around to make sure you have the right one), a twelve-sided dodecahedron with an 8.3% chance of any side showing, and, as in Stranger Things, the granddaddy of them all, a twenty-sided die, or d20, with a 5% chance of a result. Commonly, these dice are referred to as d (for die) and the number of sides. So our standard cubes are d6s, and a twelve-sided die becomes a d12.
These additional dice allow games to have a greater range of results, and provide more precision with the odds. In the case of trying to roll a 13 or more to hit Demogorgon with a fireball, Will had a 60% chance of failing to hit with his fireball if he had rolled a 1 through 12, and a 40% chance of rolling a 13 or higher to defeat Demogorgon.
With all these dice at hand, it would seem that we could model almost any number of situations. But, there is one last complication. When using a single die, there is a linear distribution of results. The chances that any single number will be rolled are the same as those for any other number. In other words, when Will rolls his twenty he has the same chance of rolling a 1 as he has of rolling a 20, or even a thirteen. If you were to graph the percentage chance of these results occurring, the graph would look like a line.
However in life, results aren’t linear. There are times when results should be weighted toward the average. Think about any large group of people. At any task, there will always be people who show a gift or exceptional skill at the task, and there will be those who have a very difficult time. Most people tend to end up lumped in the middle. The easiest way to simulate this movement toward the average is to roll multiple dice and add them together for the result. A common example of this in games is to roll two d6 (or 2d6). The most common result of those two dice will be a 7 (16.67% of the time – same as rolling a single result on d6). However, the chance of rolling the lowest number, a 2, or the highest number, a 12, is only 2.77% likely. You’ll have a 13.88% chance of having a 6 or an 8 and an 11.11% of having a 5 or a 9. Two-thirds of the time, you’ll roll between a 5 and a 9, something to consider in games like Monopoly. It’s also why Dungeons & Dragons traditionally has players roll 3d6 to generate the starting abilities of their characters, or why the damage on Will’s fireball would have been rolled on multiple dice, because it makes less physical sense that getting hit with a ball of infernal flame would be just as likely to cause negligible damage as massive damage.
Once you understand probabilities, you can manipulate them. In roleplaying games, this is the role (see what I did there?) of the game master or referee. Depending on the circumstances involved in the scene, they may add or subtract modifiers, such as a plus or minus 1 to a roll. Players may have the ability to alter probabilities as well through attributes, abilities or special items they’ve collected. Most magic weapons in Dungeons & Dragons are described through their modifiers, such as a +1 sword or the amazing +5 sword. These increase the results of the d20 die rolls in combat, adding 5% to the chances for every +1. So, if hypothetically, Will had a +5 wand of casting fireballs, he’d only need an 8 on the d20, and have a 65% chance of success instead of 40%.
So, what does this mean for the real world? Understanding probabilities and trying to model them allows us to gather information, model data and informs future decisions. While some things are cut and dry, there is nearly always a level of risk involved. We need to understand the risks to make decisions. When the kids knew they needed a 13 or more to stop Demogorgon, they understood the odds and decided to take the risk, knowing that the reward would be victory, as opposed to going the safer way with the spell of protection. Any modeling of behavior leads down a path to statistics and probabilities.
There’s also an entertainment value in probability. It gives us a wonderful feeling of success to beat the odds, whether we have any control over the outcome or not. It’s almost like magic when we roll dice, and while logically and mathematically we may know that superstitions don’t make a difference, still we have our superstitions: blowing on the dice before a roll, talking to them, keeping them in a special bag between games.
However, systems can get too complicated. We can input factors and make modifiers and adjust calculations, and while that may make the mechanics more accurate, we can lose that spark of magic—that single moment where it’s all on the line.
Let’s go back to the boys playing in Mike’s basement that night in Stranger Things. It was 1983, and they would have been playing 1st edition Advanced Dungeons & Dragons. In that system, fireball was a spell that didn’t require a roll to hit. Instead, Mike, as the Dungeon Master, would have needed to roll to see if the Demogorgon was effected by the spell, which the game modelled through something called a saving throw.
A saving throw in 1st edition was based on the hit points and magical abilities of the creature.
Being extremely powerful, Demogorgon would get a saving throw as a magic-user, and based on his hit points and magical abilities would have needed to roll a 4 or better on a d20 to save (so only a 1, 2, or 3 fails = 15% chance of failure).
Sounds grim for the good guys, but not all is lost. Fireball in 1st edition was one of the most powerful wizard spells. A saving throw against it would only act to reduce the damage taken by Demogorgon by half. Unfortunately, Demogorgon (or Prince of Demons) was one of the most powerful monsters in the game and had 200 hit points. Fireball spells did a six-sided die worth of damage per level of the caster. In order for Will’s character to have had the ability to do 200 points of damage to kill Demogorgon, he would have needed to be at least 34th level and rolled all sixes (34 x 6 = 204 points of damage), and had Demogorgon miss his saving throw. (As a side note, having a 34th level character in 1st edition Advanced Dungeons & Dragon was most unusual as most tables and stats capped out at level 20. They published the stats for Merlin and he was only 25th level!)
Still, it’s possible that Will was a 34th level wizard and that he was about to cast the mother of all fireballs, but, that’s not the only problem that the characters faced. Demogorgon had a 95% magic resistance! This would mean that even if the monster failed his save he would still have had the chance to roll a percentile die (two d10 where one represents the tens digit and one the singles digit) to see if he could simply ignore all the effects from Will’s spell. Fortunately, magic resistance was calculated against an 11th level caster in 1st edition and dropped by 5% for every level over 11th that the caster had. So, 34 – 11 = 23, and then x 5% means that Demogorgon would have lost 115% from his magic resistance, so his magic resistance would have been cancelled out by the fact that Will was the most powerful wizard in the world!
So, what are the odds that Will’s 34th level fireball kills Demogorgon and saves the day? Let’s use our new understanding of probabilities and add it up. First, Demogorgon would have needed to miss his save, that drops the odds to 15% right off the bat, and if the monster makes his save then Will would have to be a 68th level wizard for a half damage fireball (even rolling all sixes) to do 200 points of damage. But, let’s imagine that the improbable happens and Mike rolls a 1.
Demogorgon has just been blasted in the face with something like a small star. Then it would come down to Will’s damage roll. Now, this would have been awesome, because he would be rolling 34 d6! (And trust me, for a roll like this every player around that table would have pooled their dice together so he could throw them all down at once in a thunderous crash!). But, here’s the thing, the chance of Will rolling 34 straight sixes is 0.000000000000000000000000087% (0.16, the odds of getting a six on a d6 multiplied by itself 34 times). To say this is unlikely is a bit of an understatement. It is over a quadrillion times less likely than winning the lottery, and almost a quintillion times less likely than being struck and killed by an asteroid.
In reality, odds tell us that the average d6 roll will give us 3.5 (remember or discussion about rolling multiple dice and bell curves), so it is much more likely that Will’s damage roll would be around 119. It is also more probable (75% chance) that Demogorgon would make its save, cutting this number in half. So, Demogorgon would, in all probability, have taken Will’s fireball and still had something like 140 hit points. And, the battle would have raged on.
For realism this might all make sense, but it’s much less exciting and cool than having a single die roll for victory. But, then, that’s the other thing you learn from playing games: the odds tend to even out over time. Rarely does one bad roll doom you to defeat, or one good roll guarantee you victory. There will be another chance, and maybe next time you’ll face Thessalhydra, and need a 14.
So, go forth and challenge your luck. Roll the dice and try to get that natural 20 to critical and save the party. Feel the excitement when your Vulcan First Office calculates the odds of surviving another attack at 13562190123 to 1, and you manage to make through anyway. The odds may not be ever in your favor, but, like Lloyd Christmas, at least you can hope for the best.
Lloyd Christmas: Hit me with it! Just give it to me straight! I came a long way just to see you Mary, just… The least you can do is level with me. What are my chances?
Mary Swanson: Not good.
Lloyd Christmas: You mean, not good like one out of a hundred?
Mary Swanson: I’d say more like one out of a million.
Lloyd Christmas: So you’re telling me there’s a chance.