One smelly tournament player to his opponent: “I can’t believe my dice today! I should have killed WAY more stuff this turn.
Opponent, wearing an appropriate amount of deodorant, the model of all tournament players: “What do you mean?”
Smelly: “I should have killed like, I don’t know, 3 or 4 marines on average at least. Space Marines are OP!”
[Charlie, popping up from nowhere ala Adam Connover]
Charlie: Well actually, average rolls are rarely seen in reality and your math is all wrong! Hi, I’m Charlie and this is Charlie ruins mathhammer.
Close your eyes for a minute and picture this scenario. Player A controls 10 space marines and Player B also controls 10 space marines. Everything has BS3+ S4, T4, W1, AP0, D1, Sv3+, and let’s assume each marine gets 1 shot. If Player A’s space marines shot at Player B’s space marines, how many space marines would die, on average? Well, it turns out, just one. A little less than you thought? Let’s walk through how we calculate this.
We start out with 10 shots of which 66.6% hit (BS3+), leaving us with 6.66 hits. Of those hits, half wound (S4 versus T4), leaving us with 3.33 unsaved wounds. Thanks to the Sv3+, on average Player B will make 2/3 of his saves, failing 1/3 which means that 1/3 of 3.33 marines die. In other words, 1.11 space marines die on average.
Why is this important? In a game that largely revolves around killing stuff to win points or killing stuff to remove it from the board and thus take away points from your opponent, having a good idea what affect specific firepower will have is important. Turn by turn over the course of a game, you’ll have opportunities to allocate firepower from different units to different targets, or even split up shots within a unit. Knowing when you’ve diluted your firepower across too many targets can save you headaches down the road when that unit you targeted just barely survives to hold some important objective.
Smelly: “OK, so I’ll always kill one marine when shooting with ten marines, got it.”
Charlie: “Hold on there, that’s not right either. See that’s what you’ll get on average, but the average is far from guaranteed. Statistically speaking, you’re more likely to not kill one marine than to kill one marine.”
This is where variance and standard deviation comes into play. Let’s take a look at this graph, which shows the statistical probability of killing 0, 1, 2, 3, 4, or 5 marines in the above scenario.
You can see that the probability of killing 1 marine is the highest, at 38%. But the probability of any other outcome combined is far higher, at 62%. So you can see that while killing one marine is that average, based on average rolls, you’re more likely to not kill one marine (as in, kill any of 0, 2, 3, or 4) than you are to kill the one marine. Check it out for yourself, here.
In actuality and in the middle of a game, knowing the probability, exactly, of the above is probably more detailed than you’re able to quickly obtain. But it’s important to keep in mind when you roll over or under the average and start to get disparaged by what you interpret as your luck for the day. Don’t go on tilt after a few bad (below average) rolls. It’s actually fairly likely.
Smelly: “Ok, so I probably won’t roll average. How is this supposed to help me then?”
Charlie: “By setting your expectations accordingly.”
Even though most of us don’t whip out our TI89-Titanium every turn, it can be fairly useful to run through a few common scenarios when you’re making lists. By calculating the odds of the outcome of your firepower against several common target types (GEQ, MEQ, Tank, etc.) you can get a pretty good feel for what sort of firepower is needed to take out specific targets. The more thinking you can do before the game, the less you’ll have to do and the less you’ll be stressed out during the game.
Here’s a real example from my most-current game. I was playing Death Guard and Daemons against a pure Astra Militarum force. I had a Plagueburst Crawler right on top of an objective, being a general thorn in my opponent’s side. He had several units with multiple plasma-wielding guardsmen and was determined to gid rid of my PBC. He had all 14 plasmas within rapid-fire range for a total of 28 shots and most (we’ll assume all for the actual math, Veterans, etc.) was hitting on 3’s. He only managed to do 6 damage to the previously unwounded PBC and later expressed that “I thought I had enough plasma to take it out for sure!” How likely was this scenario?
It turns out, his result was just about average, slightly below. There’s about a 79% chance that he would do at least 6 wounds. But the odds of him destroying it and doing all 12 wounds with that number of shots and that BS? Only about 17%. I haven’t played Death Guard against my buddy too much, and I think he’s still getting used to having to factor in Disgustingly Resilient. As we can see, if we stop factoring in Disgustingly Resilient, his expectation of killing it is around 61% likely, doing 12 wounds is the average result, and these results much closer match his expectations.
Smelly: “So…If I want to prepare for a game, I should make sure my expectations are realistic?”
Charlie: “Exactly. By understanding the effect an ability like Disgustingly Resilient can have on the outcome, you can plan accordingly and hopefully win more games!”
Smelly: “Or, I can just play Death Guard because Disgustingly Resilient is so strong! I can do it! I was born for this!”
Charlie: [catching a whiff of Smelly] You’re not kidding!
How often do you use mathhammer? When do you find it useful?
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