|Consider your success...(puts on sunglasses)....understated - yeaaaaaaaaahhh!|
Today, I'd like to take a look at these 'real' probabilities, break them down as far as I can, then provide a few simple rules to help make better decisions on the table top.
From Wikipedia, "the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p."
In simple terms, a Binomial Distribution gives you the exact probability of a specific event, given the number of trials and the probability of each trial.
For example, I have 10 R/MG shots from a Grenadier Platoon at an assaulting US Rifle Platoon. The US Rifle platoon is Trained, but Concealed by Smoke.
What is the probability that I score exactly 5 hits?
My "specific event" is scoring 5 hits. My number of trials is 10. My probability for each trial is 3/6 or .5 or 50%.
Now, I'm not going to go into the actual math of all this - that's what calculators are for. In this case, I'm going to use StatTrek's Binomial Calculator.
StatTrek's output is nice because it not only gives me the discrete probability for our specific event (5 hits), but it also gives me the cumulative probability for Less Than 5 Hits and Greater Than Or Equal To 5 Hits.
Let's break these numbers down:
Binomial Probability: P(X = 5) - I have a 24.61% chance to score exactly 5 hits.
Cumulative Probability: P(X < 5) - I have a 37.70% chance to score less than 5 hits.
Cumulative Probability: P(X > 5) - I have a 62.30% chance to score 5 or more hits.
Now let's look at our EV calculation for the same event:
.5 (to hit) * 10 shots = 5 hits
Given our interpretation of EV, I should "expect" to score 5 hits and pin down my opponent's platoon.
However, our Binomial probability calculation shows that I only have a 62% chance to score 5 or more hits - while it is certainly greater than a 50/50 chance, we shouldn't be surprised if I fail to pin that platoon.
Riha's 80/20 Rule
I'm stealing this line from Pareto, but applying it in a different way. If I am going to risk more than 20% of my tactical strength, I want more than an 80% chance of success in my maneuver.
This balance of Risk/Reward is one of the most important parts of making sound tactical decisions. The 80/20 rule gives us an easy reference tool to make those decisions faster and more effectively.
In other words, if I'm taking some pot shots from my King Tiger at some moving infantry 34 inches away (little risk), I don't need to worry about the math. However, if I'm maneuvering to assault that bazooka laden platoon of infantry with 2 King Tigers, I want to line up my chance for success at 80% or higher (pin/smoke/etc). If I can't line my chance of success up at 80%+, then I should probably reconsider this decision.
Let's go back to our pinning example. If I want to have an 80% success rate, how many shots do I need to fire at the opposing platoon?
Thankfully, the Binomial Calculator makes it easy to plop new variables in as we please.
11 shots gives me a 72% chance to pin.
And 12 shots gives me an 80.6% chance to pin, crossing our 80/20 mark.
So when we are planning out our movement before shooting, and I really need to pin this platoon (because they are going to wreck more than 20% of my army if I don't pin them), then I should plan to fire 12 or more shots at this platoon - not the 10 suggested by a straight EV calculation.
Riha's Easy-Peasy Rule
So I know what you're all thinking: How can I learn to calculate Binomial Probabilities on the fly? (Okay, so maybe no one is actually thinking that.)
But there's an easier way to get close to 80%: Calculate your Expected Successes (EV). For every 10 dice you roll, reduce your EV by 1.
So if you fire 12 shots needing 5's, you would expect to hit (EV = 12 * .333 = 4) 4 times. But in actuality, you only have a 60% chance to score 4 or more hits. We subtract 1 from our EV and it turns out you have an 81% chance to score 3 or more hits - crossing our 80/20 line for tactical decision making.
This allows you to adjust your decision making appropriately, without having to go through all of the trouble of those binomial calculations - the buzzfeed title was right after all!
The math is strong with this article, so feel free to drop a line on the forums for an open discussion. I look forward to seeing you there.