Eric Harvey wrote:
Some design work I've been doing (a rather simplified abstraction of naval combat) put a single QE's chances of sinking the Yamato at approx 5% ...
But, when you add a few more British BBs to the mix, the math would seem to indicate that you'd need 20(!) British BBs to mathematically guarantee that the Yamato couldn't survive. ...
But you guys make some excellent points, namely that much would perhaps depend on the dueling ships relative positions, and the tactics they employed during the fight.
You need to be careful about math with probability-based statistics. Probabilities don't add up like discrete numbers do. They change based on sample size, shifting prior vs. following event constraint, and causal linkages, among other factors.
Perhaps I may give you a case to illustrate?
Let us say that you have force A, with cruisers, and force B, with destroyers. You determine, by your modelling, that the cruisers are 4 times as "lethal" to the destroyers, as the destroyers are to the cruisers. You create a simple model. It says that a cruiser will kill a destroyer with 100% of its "shots" (whatever you consider to be a quantum of firing .. a broadside, or 5 minutes of shooting, or whatever...). But the destroyers will only kill cruisers with 25% of their "shots".
Now, for the sake of simplicity in your simulation, you assume that the two sides take turns. Shots are not taken simultaneously.
If I told you that the scenario pits Force A with 10 cruisers, against Force B with 30 destroyers, and Force A (the cruisers) gets to shoot first, which side do you think will win?
Remember, the cruisers are 4X as lethal as the destroyers. There are 3X as many destroyers.
I will even grant every marginal judgement to the favor of Force A. They get first shot, and any rounding will be done to their favor.
Arithmetic would lead you to expect Force A to win. Statistics will tell you otherwise. The answer, if you follow the simulation, is that Force B (the destroyers) will win, decisively.
Observe:
Starting Balance: A = 10, B = 30 (1:3 ratio)
Turn 1:
Force A fires 10 times, and destroys 10 destroyers.
Balance: A = 10, B = 20 (1:2 ratio)
Turn 2:
Force B fires 20 times, and destroys 5 cruisers.
Balance: A = 5, B = 20 (1:4 ratio!)
Turn 3:
Force A fires 5 times, and destroys 5 destroyers.
Balance: A = 5, B = 15 (1:3 ratio)
Turn 4:
Force B fires 15 times, and (rounding to A's favor) destroys 3 cruisers
Balance: A = 2, B = 15 (1:7 ratio!)
Turn 5:
Force A fires 2 times, and destroys 2 destroyers
Balance: A = 2, B = 13 (1:6 ratio)
Turn 6:
Force B fires 13 times, and destroys all remaining cruisers (could kill 3, but only 2 remain)
Balance: A = 0, B = 13
Force A had ships that were 4X more effective. Do the arithmetic, and you would have expected a 4 to 1 exchange rate. Do the stats, and you find the exchange rate to be only 1.7 to 1, not 4 to 1.
Unfortunately it is hard to use the 1 vs. 10 scenario if you keep the rule about rounding, as stats using discrete steps (rounding to whole numbers) close to the sample size (1 Yamato) are always skewed.
But do this same simulation with 4 superbattleships that are 100% effective, versus 40 old battleships that are 5% effective. Same 20 to 1 effectiveness, versus 1 to 10 balance of forces. What do you think your results will be?
You'll end up after 8 turns with 30 surviving old battleships, and no Yamatos. 20 to 1 superiority, but only a 5 to 1 exchange rate.
As Stalin said, "sometimes quantity has a quality all it's own".